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Theorem hoidmvlelem3 40811
Description: This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
hoidmvlelem3.l 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
hoidmvlelem3.x (𝜑 → 𝑋 ∈ Fin)
hoidmvlelem3.y (𝜑 → 𝑌 ⊆ 𝑋)
hoidmvlelem3.z (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
hoidmvlelem3.w 𝑊 = (𝑌 ∪ {𝑍})
hoidmvlelem3.a (𝜑 → 𝐴:𝑊⟶ℝ)
hoidmvlelem3.b (𝜑 → 𝐵:𝑊⟶ℝ)
hoidmvlelem3.lt ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
hoidmvlelem3.f 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
hoidmvlelem3.c (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem3.j 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem3.d (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
hoidmvlelem3.k 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
hoidmvlelem3.r (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
hoidmvlelem3.h 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
hoidmvlelem3.g 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
hoidmvlelem3.e (𝜑 → 𝐸 ∈ ℝ+)
hoidmvlelem3.u 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
hoidmvlelem3.s (𝜑 → 𝑆 ∈ 𝑈)
hoidmvlelem3.sb (𝜑 → 𝑆 < (𝐵‘𝑍))
hoidmvlelem3.p 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
hoidmvlelem3.i (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
hoidmvlelem3.i2 (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
hoidmvlelem3.o 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
Assertion
Ref Expression
hoidmvlelem3 (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Distinct variable groups:   𝑥,𝑘   𝐴,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝐴,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘   𝑧,𝐴,ℎ,𝑗   𝐵,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐵,𝑐,ℎ,𝑗,𝑘,𝑥   𝐵,𝑓,𝑔   𝑢,𝐵,ℎ,𝑗   𝑧,𝐵   𝐶,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐶,𝑐   𝑢,𝐶   𝑧,𝐶   𝐷,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝐷,𝑐   𝑢,𝐷   𝑧,𝐷   𝐸,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦   𝐸,𝑐   𝑧,𝐸   𝑗,𝐹   𝐺,𝑎,𝑏,ℎ,𝑘,𝑥,𝑦   𝐺,𝑐   𝑧,𝐺   𝐻,𝑎,𝑏,𝑗,𝑘   𝑧,𝐻   𝐽,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝑔,𝐽   𝐾,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝐿,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥   𝑒,𝐿,𝑓,𝑔   𝑧,𝐿   𝑗,𝑂,𝑘   𝑃,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑃,𝑐   𝑆,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑆,𝑐   𝑢,𝑆   𝑧,𝑆   𝑢,𝑈   𝑊,𝑎,𝑏,𝑗,𝑘,𝑥   𝑊,𝑐   𝑧,𝑊   𝑌,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑌,𝑐   𝑒,𝑌,𝑓,𝑔   𝑍,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝑍,𝑐   𝑢,𝑍   𝑧,𝑍   𝜑,𝑎,𝑏,ℎ,𝑗,𝑘,𝑥,𝑦   𝜑,𝑐
Allowed substitution hints:   𝜑(𝑧,𝑢,𝑒,𝑓,𝑔)   𝐴(𝑦,𝑢,𝑐)   𝐵(𝑒)   𝐶(𝑒,𝑓,𝑔)   𝐷(𝑒,𝑓,𝑔)   𝑃(𝑧,𝑢,𝑒,𝑓,𝑔)   𝑆(𝑒,𝑓,𝑔)   𝑈(𝑥,𝑦,𝑧,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐)   𝐸(𝑢,𝑒,𝑓,𝑔,𝑗)   𝐹(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑘,𝑎,𝑏,𝑐)   𝐺(𝑢,𝑒,𝑓,𝑔,𝑗)   𝐻(𝑥,𝑦,𝑢,𝑒,𝑓,𝑔,ℎ,𝑐)   𝐽(𝑦,𝑧,𝑢,𝑒,𝑓,𝑐)   𝐾(𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,𝑐)   𝐿(𝑦,𝑢,𝑐)   𝑂(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑎,𝑏,𝑐)   𝑊(𝑦,𝑢,𝑒,𝑓,𝑔,ℎ)   𝑋(𝑥,𝑦,𝑧,𝑢,𝑒,𝑓,𝑔,ℎ,𝑗,𝑘,𝑎,𝑏,𝑐)   𝑌(𝑧,𝑢)   𝑍(𝑒,𝑓,𝑔)
Proof of Theorem hoidmvlelem3
Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1nn 11031 . . . . 5 1 ∈ ℕ
21a1i 11 . . . 4 ((𝜑 ∧ 𝑌 = ∅) → 1 ∈ ℕ)
3 0le0 11110 . . . . . 6 0 ≤ 0
43a1i 11 . . . . 5 ((𝜑 ∧ 𝑌 = ∅) → 0 ≤ 0)
5 hoidmvlelem3.g . . . . . . . 8 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))
65a1i 11 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
7 fveq2 6191 . . . . . . . . 9 (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅))
8 reseq2 5391 . . . . . . . . . 10 (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅))
9 res0 5400 . . . . . . . . . . 11 (𝐴 ↾ ∅) = ∅
109a1i 11 . . . . . . . . . 10 (𝑌 = ∅ → (𝐴 ↾ ∅) = ∅)
118, 10eqtrd 2656 . . . . . . . . 9 (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅)
12 reseq2 5391 . . . . . . . . . 10 (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅))
13 res0 5400 . . . . . . . . . . 11 (𝐵 ↾ ∅) = ∅
1413a1i 11 . . . . . . . . . 10 (𝑌 = ∅ → (𝐵 ↾ ∅) = ∅)
1512, 14eqtrd 2656 . . . . . . . . 9 (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅)
167, 11, 15oveq123d 6671 . . . . . . . 8 (𝑌 = ∅ → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
1716adantl 482 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅))
18 hoidmvlelem3.l . . . . . . . 8 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
19 f0 6086 . . . . . . . . 9 ∅:∅⟶ℝ
2019a1i 11 . . . . . . . 8 ((𝜑 ∧ 𝑌 = ∅) → ∅:∅⟶ℝ)
2118, 20, 20hoidmv0val 40797 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) = 0)
226, 17, 213eqtrd 2660 . . . . . 6 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0)
23 nfcvd 2765 . . . . . . . . . . 11 (𝜑 → Ⅎ𝑗(𝑃‘1))
24 nfv 1843 . . . . . . . . . . 11 Ⅎ𝑗𝜑
25 simpr 477 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1)
2625fveq2d 6195 . . . . . . . . . . 11 ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1))
27 1red 10055 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℝ)
28 rge0ssre 12280 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ ℝ
29 id 22 . . . . . . . . . . . . . 14 (𝜑 → 𝜑)
301a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 1 ∈ ℕ)
311elexi 3213 . . . . . . . . . . . . . . 15 1 ∈ V
32 eleq1 2689 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈ ℕ))
3332anbi2d 740 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
34 fveq2 6191 . . . . . . . . . . . . . . . . 17 (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1))
3534eleq1d 2686 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝑃‘𝑗) ∈ (0[,)+∞) ↔ (𝑃‘1) ∈ (0[,)+∞)))
3633, 35imbi12d 334 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))))
37 id 22 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ → 𝑗 ∈ ℕ)
38 ovexd 6680 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V)
39 hoidmvlelem3.p . . . . . . . . . . . . . . . . . . 19 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4039fvmpt2 6291 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4137, 38, 40syl2anc 693 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
4241adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
43 hoidmvlelem3.x . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 𝑋 ∈ Fin)
44 hoidmvlelem3.w . . . . . . . . . . . . . . . . . . . . . 22 𝑊 = (𝑌 ∪ {𝑍})
4544a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 𝑊 = (𝑌 ∪ {𝑍}))
46 hoidmvlelem3.y . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 𝑌 ⊆ 𝑋)
47 hoidmvlelem3.z . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))
4847eldifad 3586 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 𝑍 ∈ 𝑋)
49 snssi 4339 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋)
5048, 49syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑍} ⊆ 𝑋)
5146, 50unssd 3789 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋)
5245, 51eqsstrd 3639 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → 𝑊 ⊆ 𝑋)
5343, 52ssfid 8183 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑊 ∈ Fin)
54 ssun1 3776 . . . . . . . . . . . . . . . . . . . . 21 𝑌 ⊆ (𝑌 ∪ {𝑍})
5544eqcomi 2631 . . . . . . . . . . . . . . . . . . . . 21 (𝑌 ∪ {𝑍}) = 𝑊
5654, 55sseqtri 3637 . . . . . . . . . . . . . . . . . . . 20 𝑌 ⊆ 𝑊
5756a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝑌 ⊆ 𝑊)
5853, 57ssfid 8183 . . . . . . . . . . . . . . . . . 18 (𝜑 → 𝑌 ∈ Fin)
5958adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin)
60 iftrue 4092 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
6160adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
62 hoidmvlelem3.c . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
6362ffvelrnda 6359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
64 elmapi 7879 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ)
6563, 64syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ)
6654, 44sseqtr4i 3638 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑌 ⊆ 𝑊
6766a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊)
6865, 67fssresd 6071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
69 reex 10027 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ℝ ∈ V
7069a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈ V)
7153, 57ssexd 4805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → 𝑌 ∈ V)
7271adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V)
7370, 72elmapd 7871 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
7468, 73mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
7574adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
7661, 75eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
77 iffalse 4095 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
7877adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
79 0red 10041 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ)
80 hoidmvlelem3.f . . . . . . . . . . . . . . . . . . . . . . . . 25 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)
8179, 80fmptd 6385 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 𝐹:𝑌⟶ℝ)
8269a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ℝ ∈ V)
8382, 58elmapd 7871 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐹 ∈ (ℝ ↑𝑚 𝑌) ↔ 𝐹:𝑌⟶ℝ))
8481, 83mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 𝐹 ∈ (ℝ ↑𝑚 𝑌))
8584ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ (ℝ ↑𝑚 𝑌))
8678, 85eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
8776, 86pm2.61dan 832 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
88 hoidmvlelem3.j . . . . . . . . . . . . . . . . . . . 20 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
8987, 88fmptd 6385 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐽:ℕ⟶(ℝ ↑𝑚 𝑌))
9089ffvelrnda 6359 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑𝑚 𝑌))
91 elmapi 7879 . . . . . . . . . . . . . . . . . 18 ((𝐽‘𝑗) ∈ (ℝ ↑𝑚 𝑌) → (𝐽‘𝑗):𝑌⟶ℝ)
9290, 91syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
93 iftrue 4092 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
9493adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
95 hoidmvlelem3.d . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
9695ffvelrnda 6359 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊))
97 elmapi 7879 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐷‘𝑗) ∈ (ℝ ↑𝑚 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ)
9896, 97syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ)
9998, 67fssresd 6071 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
10070, 72elmapd 7871 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
10199, 100mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
102101adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
10394, 102eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
104 iffalse 4095 . . . . . . . . . . . . . . . . . . . . . . 23 (¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
105104adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹)
106105, 85eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
107103, 106pm2.61dan 832 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚 𝑌))
108 hoidmvlelem3.k . . . . . . . . . . . . . . . . . . . 20 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
109107, 108fmptd 6385 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 𝐾:ℕ⟶(ℝ ↑𝑚 𝑌))
110109ffvelrnda 6359 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑𝑚 𝑌))
111 elmapi 7879 . . . . . . . . . . . . . . . . . 18 ((𝐾‘𝑗) ∈ (ℝ ↑𝑚 𝑌) → (𝐾‘𝑗):𝑌⟶ℝ)
112110, 111syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ)
11318, 59, 92, 112hoidmvcl 40796 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞))
11442, 113eqeltrd 2701 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
11531, 36, 114vtocl 3259 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝑃‘1) ∈ (0[,)+∞))
11629, 30, 115syl2anc 693 . . . . . . . . . . . . 13 (𝜑 → (𝑃‘1) ∈ (0[,)+∞))
11728, 116sseldi 3601 . . . . . . . . . . . 12 (𝜑 → (𝑃‘1) ∈ ℝ)
118117recnd 10068 . . . . . . . . . . 11 (𝜑 → (𝑃‘1) ∈ ℂ)
11923, 24, 26, 27, 118sumsnd 39185 . . . . . . . . . 10 (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
120119adantr 481 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1))
121 fveq2 6191 . . . . . . . . . . . . 13 (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1))
122 fveq2 6191 . . . . . . . . . . . . 13 (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1))
123121, 122oveq12d 6668 . . . . . . . . . . . 12 (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
124 ovex 6678 . . . . . . . . . . . 12 ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V
125123, 39, 124fvmpt 6282 . . . . . . . . . . 11 (1 ∈ ℕ → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
1261, 125ax-mp 5 . . . . . . . . . 10 (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))
127126a1i 11 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)))
1287oveqd 6667 . . . . . . . . . . 11 (𝑌 = ∅ → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
129128adantl 482 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1)))
130121feq1d 6030 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝐽‘𝑗):𝑌⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
13133, 130imbi12d 334 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)))
13268adantr 481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)
13361feq1d 6030 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ))
134132, 133mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
13581ad2antrr 762 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ)
13678feq1d 6030 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ))
137135, 136mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
138134, 137pm2.61dan 832 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)
139 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
140 fvex 6201 . . . . . . . . . . . . . . . . . . . . 21 (𝐶‘𝑗) ∈ V
141140resex 5443 . . . . . . . . . . . . . . . . . . . 20 ((𝐶‘𝑗) ↾ 𝑌) ∈ V
14261, 141syl6eqel 2709 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
14384elexd 3214 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → 𝐹 ∈ V)
144143adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹 ∈ V)
145144adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ V)
14678, 145eqeltrd 2701 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
147142, 146pm2.61dan 832 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
14888fvmpt2 6291 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
149139, 147, 148syl2anc 693 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
150149feq1d 6030 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ))
151138, 150mpbird 247 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ)
15231, 131, 151vtocl 3259 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ)
15329, 30, 152syl2anc 693 . . . . . . . . . . . . 13 (𝜑 → (𝐽‘1):𝑌⟶ℝ)
154153adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):𝑌⟶ℝ)
155 id 22 . . . . . . . . . . . . . . 15 (𝑌 = ∅ → 𝑌 = ∅)
156155eqcomd 2628 . . . . . . . . . . . . . 14 (𝑌 = ∅ → ∅ = 𝑌)
157156feq2d 6031 . . . . . . . . . . . . 13 (𝑌 = ∅ → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
158157adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1):∅⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ))
159154, 158mpbird 247 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ)
160122feq1d 6030 . . . . . . . . . . . . . . . 16 (𝑗 = 1 → ((𝐾‘𝑗):𝑌⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
16133, 160imbi12d 334 . . . . . . . . . . . . . . 15 (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)))
16231, 161, 112vtocl 3259 . . . . . . . . . . . . . 14 ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ)
16329, 30, 162syl2anc 693 . . . . . . . . . . . . 13 (𝜑 → (𝐾‘1):𝑌⟶ℝ)
164163adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ)
165156feq2d 6031 . . . . . . . . . . . . 13 (𝑌 = ∅ → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
166165adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 = ∅) → ((𝐾‘1):∅⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ))
167164, 166mpbird 247 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ)
16818, 159, 167hoidmv0val 40797 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0)
169129, 168eqtrd 2656 . . . . . . . . 9 ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0)
170120, 127, 1693eqtrd 2660 . . . . . . . 8 ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0)
171170oveq2d 6666 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0))
172 hoidmvlelem3.e . . . . . . . . . . . 12 (𝜑 → 𝐸 ∈ ℝ+)
173172rpred 11872 . . . . . . . . . . 11 (𝜑 → 𝐸 ∈ ℝ)
17427, 173readdcld 10069 . . . . . . . . . 10 (𝜑 → (1 + 𝐸) ∈ ℝ)
175174recnd 10068 . . . . . . . . 9 (𝜑 → (1 + 𝐸) ∈ ℂ)
176175mul01d 10235 . . . . . . . 8 (𝜑 → ((1 + 𝐸) · 0) = 0)
177176adantr 481 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0)
178 eqidd 2623 . . . . . . 7 ((𝜑 ∧ 𝑌 = ∅) → 0 = 0)
179171, 177, 1783eqtrd 2660 . . . . . 6 ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0)
18022, 179breq12d 4666 . . . . 5 ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0))
1814, 180mpbird 247 . . . 4 ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
182 oveq2 6658 . . . . . . . . 9 (𝑚 = 1 → (1...𝑚) = (1...1))
1831nnzi 11401 . . . . . . . . . . 11 1 ∈ ℤ
184 fzsn 12383 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
185183, 184ax-mp 5 . . . . . . . . . 10 (1...1) = {1}
186185a1i 11 . . . . . . . . 9 (𝑚 = 1 → (1...1) = {1})
187182, 186eqtrd 2656 . . . . . . . 8 (𝑚 = 1 → (1...𝑚) = {1})
188187sumeq1d 14431 . . . . . . 7 (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗))
189188oveq2d 6666 . . . . . 6 (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))
190189breq2d 4665 . . . . 5 (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))))
191190rspcev 3309 . . . 4 ((1 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
1922, 181, 191syl2anc 693 . . 3 ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
193 simpl 473 . . . 4 ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑)
194 neqne 2802 . . . . 5 (¬ 𝑌 = ∅ → 𝑌 ≠ ∅)
195194adantl 482 . . . 4 ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅)
196 nfv 1843 . . . . . 6 Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅)
197183a1i 11 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈ ℤ)
198 nnuz 11723 . . . . . 6 ℕ = (ℤ≥‘1)
199114adantlr 751 . . . . . 6 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞))
200 hoidmvlelem3.a . . . . . . . . . . . 12 (𝜑 → 𝐴:𝑊⟶ℝ)
20166a1i 11 . . . . . . . . . . . 12 (𝜑 → 𝑌 ⊆ 𝑊)
202200, 201fssresd 6071 . . . . . . . . . . 11 (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
203 hoidmvlelem3.b . . . . . . . . . . . 12 (𝜑 → 𝐵:𝑊⟶ℝ)
204203, 201fssresd 6071 . . . . . . . . . . 11 (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
20518, 58, 202, 204hoidmvcl 40796 . . . . . . . . . 10 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞))
20628, 205sseldi 3601 . . . . . . . . 9 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ)
2075, 206syl5eqel 2705 . . . . . . . 8 (𝜑 → 𝐺 ∈ ℝ)
208 0red 10041 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
209 1rp 11836 . . . . . . . . . . . . 13 1 ∈ ℝ+
210209a1i 11 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ+)
211210, 172jca 554 . . . . . . . . . . 11 (𝜑 → (1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+))
212 rpaddcl 11854 . . . . . . . . . . 11 ((1 ∈ ℝ+ ∧ 𝐸 ∈ ℝ+) → (1 + 𝐸) ∈ ℝ+)
213211, 212syl 17 . . . . . . . . . 10 (𝜑 → (1 + 𝐸) ∈ ℝ+)
214 rpgt0 11844 . . . . . . . . . 10 ((1 + 𝐸) ∈ ℝ+ → 0 < (1 + 𝐸))
215213, 214syl 17 . . . . . . . . 9 (𝜑 → 0 < (1 + 𝐸))
216208, 215gtned 10172 . . . . . . . 8 (𝜑 → (1 + 𝐸) ≠ 0)
217207, 174, 216redivcld 10853 . . . . . . 7 (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ)
218217adantr 481 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
219217ltpnfd 11955 . . . . . . . . . 10 (𝜑 → (𝐺 / (1 + 𝐸)) < +∞)
220219adantr 481 . . . . . . . . 9 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < +∞)
221 id 22 . . . . . . . . . . 11 ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
222221eqcomd 2628 . . . . . . . . . 10 ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
223222adantl 482 . . . . . . . . 9 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → +∞ = (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
224220, 223breqtrd 4679 . . . . . . . 8 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
225224adantlr 751 . . . . . . 7 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
226 simpl 473 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝜑 ∧ 𝑌 ≠ ∅))
227 simpr 477 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)
228 nnex 11026 . . . . . . . . . . . 12 ℕ ∈ V
229228a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈ V)
230 icossicc 12260 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ (0[,]+∞)
231230, 114sseldi 3601 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞))
232 eqid 2622 . . . . . . . . . . . . 13 (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗))
233231, 232fmptd 6385 . . . . . . . . . . . 12 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
234233adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞))
235229, 234sge0repnf 40603 . . . . . . . . . 10 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞))
236227, 235mpbird 247 . . . . . . . . 9 ((𝜑 ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
237236adantlr 751 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
238218adantr 481 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
239207adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
240239adantlr 751 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ)
241 simpr 477 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ)
24227, 172ltaddrpd 11905 . . . . . . . . . . . 12 (𝜑 → 1 < (1 + 𝐸))
243242adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → 1 < (1 + 𝐸))
24458adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ Fin)
245 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ≠ ∅)
246202adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐴 ↾ 𝑌):𝑌⟶ℝ)
247204adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐵 ↾ 𝑌):𝑌⟶ℝ)
24818, 244, 245, 246, 247hoidmvn0val 40798 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
2495a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
250 fvres 6207 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
251 fvres 6207 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
252250, 251oveq12d 6668 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
253252fveq2d 6195 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
254253adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
255200adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ)
256 elun1 3780 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
257256, 44syl6eleqr 2712 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊)
258257adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
259255, 258ffvelrnd 6360 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ)
260203adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ)
261260, 258ffvelrnd 6360 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ)
262 volico 40200 . . . . . . . . . . . . . . . . . . 19 (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
263259, 261, 262syl2anc 693 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
264 hoidmvlelem3.lt . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))
265258, 264syldan 487 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘))
266265iftrued 4094 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
267254, 263, 2663eqtrd 2660 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
268267prodeq2dv 14653 . . . . . . . . . . . . . . . 16 (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
269268eqcomd 2628 . . . . . . . . . . . . . . 15 (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
270269adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))))
271248, 249, 2703eqtr4d 2666 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)))
272 difrp 11868 . . . . . . . . . . . . . . . . 17 (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
273259, 261, 272syl2anc 693 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+))
274265, 273mpbid 222 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
27558, 274fprodrpcl 14686 . . . . . . . . . . . . . 14 (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
276275adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ+)
277271, 276eqeltrd 2701 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ ℝ+)
278213adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → (1 + 𝐸) ∈ ℝ+)
279277, 278ltdivgt1 39572 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺))
280243, 279mpbid 222 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < 𝐺)
281280adantr 481 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < 𝐺)
282 hoidmvlelem3.i2 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
283282adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
284 fvexd 6203 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑥‘𝑘) ∈ V)
285 hoidmvlelem3.s . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → 𝑆 ∈ 𝑈)
286285elexd 3214 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → 𝑆 ∈ V)
287284, 286ifcld 4131 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
288287ralrimivw 2967 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
289288adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
290 eqid 2622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
291290fnmpt 6020 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
292289, 291syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)
293 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
294 mptexg 6484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑊 ∈ Fin → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
29553, 294syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
296295adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V)
297 hoidmvlelem3.o . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
298297fvmpt2 6291 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
299293, 296, 298syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
300299fneq1d 5981 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ↔ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊))
301292, 300mpbird 247 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊)
302 nfv 1843 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘𝜑
303 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘𝑥
304 nfixp1 7928 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
305303, 304nfel 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
306302, 305nfan 1828 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
307299fveq1d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
308307adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
309 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊)
310287adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
311290fvmpt2 6291 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∈ 𝑊 ∧ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
312309, 310, 311syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
313312adantlr 751 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
314308, 313eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
315 iftrue 4092 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
316315adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
317 vex 3203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑥 ∈ V
318317elixp 7915 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
319318simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
320319adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
321 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌)
322 rspa 2930 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
323320, 321, 322syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
324323ad4ant24 1298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
325316, 324eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
326 snidg 4206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍})
32747, 326syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → 𝑍 ∈ {𝑍})
328 elun2 3781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍}))
329327, 328syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍}))
33055a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊)
331329, 330eleqtrd 2703 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → 𝑍 ∈ 𝑊)
332200, 331ffvelrnd 6360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝐴‘𝑍) ∈ ℝ)
333332rexrd 10089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐴‘𝑍) ∈ ℝ*)
334203, 331ffvelrnd 6360 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝐵‘𝑍) ∈ ℝ)
335334rexrd 10089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐵‘𝑍) ∈ ℝ*)
336 iccssxr 12256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆ ℝ*
337 hoidmvlelem3.u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}
338 ssrab2 3687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
339337, 338eqsstri 3635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍))
340339, 285sseldi 3601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)))
341336, 340sseldi 3601 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → 𝑆 ∈ ℝ*)
342 iccgelb 12230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆)
343333, 335, 340, 342syl3anc 1326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (𝐴‘𝑍) ≤ 𝑆)
344 hoidmvlelem3.sb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → 𝑆 < (𝐵‘𝑍))
345333, 335, 341, 343, 344elicod 12224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
346345ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
347 iffalse 4095 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ 𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
348347adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆)
34944eleq2i 2693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍}))
350349biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 ∈ 𝑊 → 𝑘 ∈ (𝑌 ∪ {𝑍}))
351350adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍}))
352 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌)
353 elunnel1 3754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
354351, 352, 353syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍})
355 elsni 4194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 ∈ {𝑍} → 𝑘 = 𝑍)
356354, 355syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍)
357 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍))
358 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍))
359357, 358oveq12d 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
360356, 359syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
361360adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍)))
362348, 361eleq12d 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → (if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))))
363346, 362mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
364363adantllr 755 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
365325, 364pm2.61dan 832 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
366314, 365eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
367366ex 450 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
368306, 367ralrimi 2957 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
369301, 368jca 554 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
370 fvex 6201 . . . . . . . . . . . . . . . . . . . . . 22 (𝑂‘𝑥) ∈ V
371370elixp 7915 . . . . . . . . . . . . . . . . . . . . 21 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))))
372369, 371sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
373283, 372sseldd 3604 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
374 eliun 4524 . . . . . . . . . . . . . . . . . . 19 ((𝑂‘𝑥) ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
375373, 374sylib 208 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
376 ixpfn 7914 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌)
377376adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌)
378377ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 Fn 𝑌)
379 nfv 1843 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘 𝑗 ∈ ℕ
380306, 379nfan 1828 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ)
381 nfcv 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘(𝑂‘𝑥)
382 nfixp1 7928 . . . . . . . . . . . . . . . . . . . . . . . . 25 Ⅎ𝑘X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
383381, 382nfel 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))
384380, 383nfan 1828 . . . . . . . . . . . . . . . . . . . . . . 23 Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
3853073adant3 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘))
386287adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V)
387258, 386, 311syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
3883873adant2 1080 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))
3893153ad2ant3 1084 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘))
390385, 388, 3893eqtrrd 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
391390ad5ant125 1312 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘))
392 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
393370elixp 7915 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
394392, 393sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))
395394simprd 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
396257adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊)
397 rspa 2930 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
398395, 396, 397syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
399398adantll 750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
400391, 399eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
40129ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝜑)
40237ad2antlr 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑗 ∈ ℕ)
403299fveq1d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍))
404 eqidd 2623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))
405 eleq1 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌))
406 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍))
407405, 406ifbieq1d 4109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
408407adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
409 fvexd 6203 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑 → (𝑥‘𝑍) ∈ V)
410409, 286ifcld 4131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V)
411404, 408, 331, 410fvmptd 6288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
412411adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆))
41347eldifbd 3587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → ¬ 𝑍 ∈ 𝑌)
414413iffalsed 4097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
415414adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆)
416403, 412, 4153eqtrrd 2661 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
417416ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍))
418401, 331syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑍 ∈ 𝑊)
419393simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
420419adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
421 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍))
422 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍))
423 fveq2 6191 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍))
424422, 423oveq12d 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
425421, 424eleq12d 2695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))))
426425rspcva 3307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑍 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
427418, 420, 426syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
428417, 427eqeltrd 2701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))
4291493adant3 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))
430603ad2ant3 1084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌))
431429, 430eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌))
432431fveq1d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
433401, 402, 428, 432syl3anc 1326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
434433adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘))
435 fvres 6207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
436435adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
437434, 436eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘))
438107elexd 3214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V)
439108fvmpt2 6291 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
440139, 438, 439syl2anc 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
4414403adant3 1081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))
442933ad2ant3 1084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌))
443441, 442eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌))
444443fveq1d 6193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
445401, 402, 428, 444syl3anc 1326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
446445adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘))
447 fvres 6207 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
448447adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
449446, 448eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘))
450437, 449oveq12d 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))
451450eqcomd 2628 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
452400, 451eleqtrd 2703 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
453452ex 450 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑌 → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
454384, 453ralrimi 2957 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
455378, 454jca 554 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
456317elixp 7915 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
457455, 456sylibr 224 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
458457ex 450 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) → ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
459458reximdva 3017 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
460375, 459mpd 15 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
461 eliun 4524 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
462460, 461sylibr 224 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
463462ralrimiva 2966 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
464 dfss3 3592 . . . . . . . . . . . . . . 15 (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
465463, 464sylibr 224 . . . . . . . . . . . . . 14 (𝜑 → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
466 ovexd 6680 . . . . . . . . . . . . . . . . 17 (𝜑 → (ℝ ↑𝑚 𝑌) ∈ V)
467228a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ ∈ V)
468466, 467elmapd 7871 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ↔ 𝐾:ℕ⟶(ℝ ↑𝑚 𝑌)))
469109, 468mpbird 247 . . . . . . . . . . . . . . 15 (𝜑 → 𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ))
470466, 467elmapd 7871 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ↔ 𝐽:ℕ⟶(ℝ ↑𝑚 𝑌)))
47189, 470mpbird 247 . . . . . . . . . . . . . . . 16 (𝜑 → 𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ))
47282, 71elmapd 7871 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ))
473204, 472mpbird 247 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
47482, 71elmapd 7871 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ))
475202, 474mpbird 247 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌))
476 hoidmvlelem3.i . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
477 fveq1 6190 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
478477adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘))
479250adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘))
480478, 479eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘))
481480oveq1d 6665 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
482481ixpeq2dva 7923 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)))
483482sseq1d 3632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (𝐴 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
484 oveq1 6657 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓))
485484breq1d 4663 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑒 = (𝐴 ↾ 𝑌) → ((𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
486483, 485imbi12d 334 . . . . . . . . . . . . . . . . . . . . . 22 (𝑒 = (𝐴 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
487486ralbidv 2986 . . . . . . . . . . . . . . . . . . . . 21 (𝑒 = (𝐴 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
488487ralbidv 2986 . . . . . . . . . . . . . . . . . . . 20 (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
489488ralbidv 2986 . . . . . . . . . . . . . . . . . . 19 (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
490489rspcva 3307 . . . . . . . . . . . . . . . . . 18 (((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ∧ ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
491475, 476, 490syl2anc 693 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
492 fveq1 6190 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
493492adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘))
494251adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘))
495493, 494eqtrd 2656 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘))
496495oveq2d 6666 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
497496ixpeq2dva 7923 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
498497sseq1d 3632 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝐵 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
499 oveq2 6658 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)))
500499breq1d 4663 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (𝐵 ↾ 𝑌) → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
501498, 500imbi12d 334 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (𝐵 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
502501ralbidv 2986 . . . . . . . . . . . . . . . . . . 19 (𝑓 = (𝐵 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
503502ralbidv 2986 . . . . . . . . . . . . . . . . . 18 (𝑓 = (𝐵 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
504503rspcva 3307 . . . . . . . . . . . . . . . . 17 (((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚 𝑌) ∧ ∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
505473, 491, 504syl2anc 693 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
506 fveq1 6190 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗))
507506fveq1d 6193 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘))
508507oveq1d 6665 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
509508ixpeq2dv 7924 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
510509iuneq2d 4547 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝐽 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))
511510sseq2d 3633 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝐽 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))))
512506oveq1d 6665 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))
513512mpteq2dv 4745 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))
514513fveq2d 6195 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = 𝐽 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))
515514breq2d 4665 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝐽 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
516511, 515imbi12d 334 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝐽 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
517516ralbidv 2986 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝐽 → (∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))))
518517rspcva 3307 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ∧ ∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
519471, 505, 518syl2anc 693 . . . . . . . . . . . . . . 15 (𝜑 → ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))
520 fveq1 6190 . . . . . . . . . . . . . . . . . . . . . 22 (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗))
521520fveq1d 6193 . . . . . . . . . . . . . . . . . . . . 21 (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘))
522521oveq2d 6666 . . . . . . . . . . . . . . . . . . . 20 (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
523522ixpeq2dv 7924 . . . . . . . . . . . . . . . . . . 19 (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
524523iuneq2d 4547 . . . . . . . . . . . . . . . . . 18 (ℎ = 𝐾 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))
525524sseq2d 3633 . . . . . . . . . . . . . . . . 17 (ℎ = 𝐾 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))))
526520oveq2d 6666 . . . . . . . . . . . . . . . . . . . 20 (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
527526mpteq2dv 4745 . . . . . . . . . . . . . . . . . . 19 (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
528527fveq2d 6195 . . . . . . . . . . . . . . . . . 18 (ℎ = 𝐾 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
529528breq2d 4665 . . . . . . . . . . . . . . . . 17 (ℎ = 𝐾 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
530525, 529imbi12d 334 . . . . . . . . . . . . . . . 16 (ℎ = 𝐾 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))))
531530rspcva 3307 . . . . . . . . . . . . . . 15 ((𝐾 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ) ∧ ∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
532469, 519, 531syl2anc 693 . . . . . . . . . . . . . 14 (𝜑 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
533465, 532mpd 15 . . . . . . . . . . . . 13 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
534 idd 24 . . . . . . . . . . . . 13 (𝜑 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
535533, 534mpd 15 . . . . . . . . . . . 12 (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
536535adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
53741adantl 482 . . . . . . . . . . . . . 14 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))
538537mpteq2dva 4744 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))
539538fveq2d 6195 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑌 ≠ ∅) → (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))
540249, 539breq12d 4666 . . . . . . . . . . 11 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))
541536, 540mpbird 247 . . . . . . . . . 10 ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
542541adantr 481 . . . . . . . . 9 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
543238, 240, 241, 281, 542ltletrd 10197 . . . . . . . 8 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
544226, 237, 543syl2anc 693 . . . . . . 7 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬ (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
545225, 544pm2.61dan 832 . . . . . 6 ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < (Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))))
546196, 197, 198, 199, 218, 545sge0uzfsumgt 40661 . . . . 5 ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
547217adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ)
548 fzfid 12772 . . . . . . . . . . . . 13 (𝜑 → (1...𝑚) ∈ Fin)
549 simpl 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑)
550 elfznn 12370 . . . . . . . . . . . . . . 15 (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ)
551550adantl 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ)
55228, 114sseldi 3601 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ)
553549, 551, 552syl2anc 693 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ)
554548, 553fsumrecl 14465 . . . . . . . . . . . 12 (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
555554adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ)
556 simpr 477 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
557547, 555, 556ltled 10185 . . . . . . . . . 10 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))
558207adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ)
559213adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈ ℝ+)
560558, 555, 559ledivmuld 11925 . . . . . . . . . 10 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
561557, 560mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
562561ex 450 . . . . . . . 8 (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
563562adantr 481 . . . . . . 7 ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
564563adantlr 751 . . . . . 6 (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
565564reximdva 3017 . . . . 5 ((𝜑 ∧ 𝑌 ≠ ∅) → (∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))))
566546, 565mpd 15 . . . 4 ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
567193, 195, 566syl2anc 693 . . 3 ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
568192, 567pm2.61dan 832 . 2 (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
569433ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin)
570463ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋)
571473ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌))
5722003ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ)
5732033ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ)
574623ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))
575 eqid 2622 . . . . 5 (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0)
576 eqid 2622 . . . . 5 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
577953ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))
578 eqid 2622 . . . . 5 (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
579 fveq2 6191 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗))
580 fveq2 6191 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗))
581579, 580oveq12d 6668 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
582581cbvmptv 4750 . . . . . . . 8 (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))
583582fveq2i 6194 . . . . . . 7 (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))
584 hoidmvlelem3.r . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)
585583, 584syl5eqel 2705 . . . . . 6 (𝜑 → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
5865853ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ)
587 hoidmvlelem3.h . . . . . 6 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))
588 eleq1 2689 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌))
589 fveq2 6191 . . . . . . . . . 10 (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖))
590589breq1d 4663 . . . . . . . . . . 11 (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥))
591590, 589ifbieq1d 4109 . . . . . . . . . 10 (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))
592588, 589, 591ifbieq12d 4113 . . . . . . . . 9 (𝑗 = 𝑖 → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))
593592cbvmptv 4750 . . . . . . . 8 (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))
594593mpteq2i 4741 . . . . . . 7 (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))))
595594mpteq2i 4741 . . . . . 6 (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
596587, 595eqtri 2644 . . . . 5 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))))
5971723ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈ ℝ+)
598 fveq2 6191 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖))
599 fveq2 6191 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖))
600599fveq2d 6195 . . . . . . . . . . . . 13 (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖)))
601598, 600oveq12d 6668 . . . . . . . . . . . 12 (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
602601cbvmptv 4750 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))
603602fveq2i 6194 . . . . . . . . . 10 (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) = (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))
604603oveq2i 6661 . . . . . . . . 9 ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))
605604breq2i 4661 . . . . . . . 8 ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))))
606605a1i 11 . . . . . . 7 (𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))))
607606rabbiia 3185 . . . . . 6 {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
608337, 607eqtri 2644 . . . . 5 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))}
6092853ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈)
6103443ad2ant1 1082 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍))
611 eqid 2622 . . . . 5 (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) = (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
612 simp2 1062 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ)
613 id 22 . . . . . . . 8 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))
614 fveq2 6191 . . . . . . . . . . 11 (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖))
615614cbvsumv 14426 . . . . . . . . . 10 Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)
616615oveq2i 6661 . . . . . . . . 9 ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))
617616a1i 11 . . . . . . . 8 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
618613, 617breqtrd 4679 . . . . . . 7 (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
6196183ad2ant3 1084 . . . . . 6 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)))
620 simpl 473 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑)
621 elfznn 12370 . . . . . . . . . . 11 (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ)
622621adantl 482 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ)
623 eleq1 2689 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ))
624 fveq2 6191 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖))
625 fveq2 6191 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
626624, 625oveq12d 6668 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
627614, 626eqeq12d 2637 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))
628623, 627imbi12d 334 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → ((𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) ↔ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))))
629628, 41chvarv 2263 . . . . . . . . . . . . 13 (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
630629adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))
631623anbi2d 740 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ)))
632598fveq1d 6193 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
633599fveq1d 6193 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
634632, 633oveq12d 6668 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
635634eleq2d 2687 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
636598reseq1d 5395 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌))
637635, 636ifbieq1d 4109 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
638624, 637eqeq12d 2637 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)))
639631, 638imbi12d 334 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))))
640639, 149chvarv 2263 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
641599reseq1d 5395 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌))
642635, 641ifbieq1d 4109 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
643625, 642eqeq12d 2637 . . . . . . . . . . . . . . 15 (𝑗 = 𝑖 → ((𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
644631, 643imbi12d 334 . . . . . . . . . . . . . 14 (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))))
645644, 440chvarv 2263 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
646640, 645oveq12d 6668 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
647630, 646eqtrd 2656 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
648 simpr 477 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ)
649 ovexd 6680 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V)
650611fvmpt2 6291 . . . . . . . . . . . . 13 ((𝑖 ∈ ℕ ∧ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
651648, 649, 650syl2anc 693 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))
652 fvex 6201 . . . . . . . . . . . . . . . . . . 19 (𝐶‘𝑖) ∈ V
653652resex 5443 . . . . . . . . . . . . . . . . . 18 ((𝐶‘𝑖) ↾ 𝑌) ∈ V
654653a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V)
65580, 143syl5eqelr 2706 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V)
656654, 655ifcld 4131 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
657656adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
658576fvmpt2 6291 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
659648, 657, 658syl2anc 693 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
66080eqcomi 2631 . . . . . . . . . . . . . . . 16 (𝑦 ∈ 𝑌 ↦ 0) = 𝐹
661 ifeq2 4091 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ 𝑌 ↦ 0) = 𝐹 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
662660, 661ax-mp 5 . . . . . . . . . . . . . . 15 if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)
663662a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
664659, 663eqtrd 2656 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))
665 fvex 6201 . . . . . . . . . . . . . . . . . . 19 (𝐷‘𝑖) ∈ V
666665resex 5443 . . . . . . . . . . . . . . . . . 18 ((𝐷‘𝑖) ↾ 𝑌) ∈ V
667666a1i 11 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V)
668667, 655ifcld 4131 . . . . . . . . . . . . . . . 16 (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
669668adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V)
670578fvmpt2 6291 . . . . . . . . . . . . . . 15 ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
671648, 669, 670syl2anc 693 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))
672 biid 251 . . . . . . . . . . . . . . . 16 (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
673672, 660ifbieq2i 4110 . . . . . . . . . . . . . . 15 if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)
674673a1i 11 . . . . . . . . . . . . . 14 ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
675671, 674eqtrd 2656 . . . . . . . . . . . . 13 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))
676664, 675oveq12d 6668 . . . . . . . . . . . 12 ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
677651, 676eqtrd 2656 . . . . . . . . . . 11 ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))
678647, 677eqtr4d 2659 . . . . . . . . . 10 ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
679620, 622, 678syl2anc 693 . . . . . . . . 9 ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
6806793ad2antl1 1223 . . . . . . . 8 (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
681680sumeq2dv 14433 . . . . . . 7 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖) = Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))
682681oveq2d 6666 . . . . . 6 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)))
683619, 682breqtrd 4679 . . . . 5 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)))
684 fveq2 6191 . . . . . . . 8 (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ))
685684fveq1d 6193 . . . . . . 7 (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍))
686685cbvmptv 4750 . . . . . 6 (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍))
687686rneqi 5352 . . . . 5 ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍))
688 fveq2 6191 . . . . . . . . . . . 12 (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖))
689688fveq1d 6193 . . . . . . . . . . 11 (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍))
690 fveq2 6191 . . . . . . . . . . . 12 (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖))
691690fveq1d 6193 . . . . . . . . . . 11 (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍))
692689, 691oveq12d 6668 . . . . . . . . . 10 (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))
693692eleq2d 2687 . . . . . . . . 9 (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))))
694693cbvrabv 3199 . . . . . . . 8 {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))}
695694mpteq1i 4739 . . . . . . 7 (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))
696695rneqi 5352 . . . . . 6 ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))
697696uneq2i 3764 . . . . 5 ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))) = ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)))
698 eqid 2622 . . . . 5 inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < ) = inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < )
69918, 569, 570, 571, 44, 572, 573, 574, 575, 576, 577, 578, 586, 596, 5, 597, 608, 609, 610, 611, 612, 683, 687, 697, 698hoidmvlelem2 40810 . . . 4 ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
7006993exp 1264 . . 3 (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)))
701700rexlimdv 3030 . 2 (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))
702568, 701mpd 15 1 (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ↑𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  infcinf 8347  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  +∞cpnf 10071  ℝ*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ℝ+crp 11832  [,)cico 12177  [,]cicc 12178  ...cfz 12326  Σcsu 14416  ∏cprod 14635  volcvol 23232  Σ^csumge0 40579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580
This theorem is referenced by:  hoidmvlelem4  40812
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