Theorem poimirlem30 33439

Description: Lemma for poimir 33442 combining poimirlem29 33438 with bwth 21213. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimir.i	⊢ 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
poimir.r	⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
poimir.1	⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))
poimirlem30.x	⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘𝑓 + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)
poimirlem30.2	⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
poimirlem30.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
poimirlem30.4	⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)

Assertion

Ref	Expression
poimirlem30	⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Distinct variable groups:   𝑓,𝑗,𝑘,𝑛,𝑧   𝜑,𝑗,𝑛   𝑗,𝐹,𝑛   𝑗,𝑁,𝑛   𝜑,𝑘   𝑓,𝑁,𝑘   𝜑,𝑧   𝑓,𝐹,𝑘,𝑧   𝑧,𝑁   𝑗,𝑐,𝑘,𝑛,𝑟,𝑣,𝑧,𝜑   𝑓,𝑐,𝐹,𝑟,𝑣   𝐺,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝐼,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝑁,𝑐,𝑟,𝑣   𝑅,𝑐,𝑓,𝑗,𝑘,𝑛,𝑟,𝑣,𝑧   𝑋,𝑐,𝑓,𝑟,𝑣,𝑧

Allowed substitution hints:   𝜑(𝑓)   𝑋(𝑗,𝑘,𝑛)

Proof of Theorem poimirlem30

Dummy variables 𝑖 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfzonn0 12512	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℕ0)
2	1	nn0red 11352	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ∈ ℝ)
3		nndivre 11056	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
4	2, 3	sylan 488	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ ℝ)
5		elfzole1 12478	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 0 ≤ 𝑖)
6	2, 5	jca 554	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (0..^𝑘) → (𝑖 ∈ ℝ ∧ 0 ≤ 𝑖))
7		nnrp 11842	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
8	7	rpregt0d 11878	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
9		divge0 10892	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 0 ≤ 𝑖) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝑖 / 𝑘))
10	6, 8, 9	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝑖 / 𝑘))
11		elfzo0le 12511	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (0..^𝑘) → 𝑖 ≤ 𝑘)
12	11	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ≤ 𝑘)
13	2	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑖 ∈ ℝ)
14		1red 10055	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
15	7	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
16	13, 14, 15	ledivmuld 11925	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ (𝑘 · 1)))
17		nncn 11028	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
18	17	mulid1d 10057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑘 · 1) = 𝑘)
19	18	breq2d 4665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
20	19	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 ≤ (𝑘 · 1) ↔ 𝑖 ≤ 𝑘))
21	16, 20	bitrd 268	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → ((𝑖 / 𝑘) ≤ 1 ↔ 𝑖 ≤ 𝑘))
22	12, 21	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ≤ 1)
23		0re 10040	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℝ
24		1re 10039	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
25	23, 24	elicc2i 12239	. . . . . . . . . . . . . 14 ⊢ ((𝑖 / 𝑘) ∈ (0[,]1) ↔ ((𝑖 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝑖 / 𝑘) ∧ (𝑖 / 𝑘) ≤ 1))
26	4, 10, 22, 25	syl3anbrc 1246	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (0..^𝑘) ∧ 𝑘 ∈ ℕ) → (𝑖 / 𝑘) ∈ (0[,]1))
27	26	ancoms 469	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑖 / 𝑘) ∈ (0[,]1))
28		elsni 4194	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ {𝑘} → 𝑗 = 𝑘)
29	28	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) = (𝑖 / 𝑘))
30	29	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ {𝑘} → ((𝑖 / 𝑗) ∈ (0[,]1) ↔ (𝑖 / 𝑘) ∈ (0[,]1)))
31	27, 30	syl5ibrcom 237	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ∧ 𝑖 ∈ (0..^𝑘)) → (𝑗 ∈ {𝑘} → (𝑖 / 𝑗) ∈ (0[,]1)))
32	31	impr 649	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℕ ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
33	32	adantll 750	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ (𝑖 ∈ (0..^𝑘) ∧ 𝑗 ∈ {𝑘})) → (𝑖 / 𝑗) ∈ (0[,]1))
34		poimirlem30.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
35	34	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
36		xp1st 7198	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑘) ∈ ((ℕ0 ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(𝐺‘𝑘)) ∈ (ℕ0 ↑𝑚 (1...𝑁)))
37		elmapfn 7880	. . . . . . . . . . 11 ⊢ ((1st ‘(𝐺‘𝑘)) ∈ (ℕ0 ↑𝑚 (1...𝑁)) → (1st ‘(𝐺‘𝑘)) Fn (1...𝑁))
38	35, 36, 37	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)) Fn (1...𝑁))
39		poimirlem30.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))
40		df-f 5892	. . . . . . . . . 10 ⊢ ((1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘) ↔ ((1st ‘(𝐺‘𝑘)) Fn (1...𝑁) ∧ ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘)))
41	38, 39, 40	sylanbrc 698	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1st ‘(𝐺‘𝑘)):(1...𝑁)⟶(0..^𝑘))
42		vex 3203	. . . . . . . . . . 11 ⊢ 𝑘 ∈ V
43	42	fconst 6091	. . . . . . . . . 10 ⊢ ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘}
44	43	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1...𝑁) × {𝑘}):(1...𝑁)⟶{𝑘})
45		fzfid 12772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑁) ∈ Fin)
46		inidm 3822	. . . . . . . . 9 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
47	33, 41, 44, 45, 45, 46	off 6912	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
48		poimir.i	. . . . . . . . . 10 ⊢ 𝐼 = ((0[,]1) ↑𝑚 (1...𝑁))
49	48	eleq2i 2693	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)))
50		ovex 6678	. . . . . . . . . 10 ⊢ (0[,]1) ∈ V
51		ovex 6678	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
52	50, 51	elmap 7886	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ ((0[,]1) ↑𝑚 (1...𝑁)) ↔ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
53	49, 52	bitri 264	. . . . . . . 8 ⊢ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼 ↔ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})):(1...𝑁)⟶(0[,]1))
54	47, 53	sylibr 224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ 𝐼)
55		eqid 2622	. . . . . . 7 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
56	54, 55	fmptd 6385	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼)
57		frn 6053	. . . . . 6 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))):ℕ⟶𝐼 → ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
58	56, 57	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼)
59		ominf 8172	. . . . . . 7 ⊢ ¬ ω ∈ Fin
60		nnenom 12779	. . . . . . . . 9 ⊢ ℕ ≈ ω
61		enfi 8176	. . . . . . . . 9 ⊢ (ℕ ≈ ω → (ℕ ∈ Fin ↔ ω ∈ Fin))
62	60, 61	ax-mp 5	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ω ∈ Fin)
63		iunid 4575	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))){𝑐} = ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
64	63	imaeq2i 5464	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))){𝑐}) = (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
65		imaiun 6503	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))){𝑐}) = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐})
66		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ V
67	66, 55	fnmpti 6022	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) Fn ℕ
68		dffn3 6054	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) Fn ℕ ↔ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
69	67, 68	mpbi 220	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
70		fimacnv 6347	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))):ℕ⟶ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) = ℕ)
71	69, 70	ax-mp 5	. . . . . . . . . 10 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) = ℕ
72	64, 65, 71	3eqtr3ri 2653	. . . . . . . . 9 ⊢ ℕ = ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐})
73	72	eleq1i 2692	. . . . . . . 8 ⊢ (ℕ ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
74	62, 73	bitr3i 266	. . . . . . 7 ⊢ (ω ∈ Fin ↔ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
75	59, 74	mtbi 312	. . . . . 6 ⊢ ¬ ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin
76		ralnex 2992	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
77	76	rexbii 3041	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
78		rexnal 2995	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ℕ ¬ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
79	77, 78	bitri 264	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
80	79	ralbii 2980	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
81		ralnex 2992	. . . . . . . . 9 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ¬ ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
82	80, 81	bitri 264	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
83		nnuz 11723	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ≥‘1)
84		elnnuz 11724	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ ↔ 𝑖 ∈ (ℤ≥‘1))
85		fzouzsplit 12503	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
86	84, 85	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘1) = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
87	83, 86	syl5eq 2668	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ℕ = ((1..^𝑖) ∪ (ℤ≥‘𝑖)))
88	87	difeq1d 3727	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)))
89		uncom 3757	. . . . . . . . . . . . . . . 16 ⊢ ((1..^𝑖) ∪ (ℤ≥‘𝑖)) = ((ℤ≥‘𝑖) ∪ (1..^𝑖))
90	89	difeq1i 3724	. . . . . . . . . . . . . . 15 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖))
91		difun2 4048	. . . . . . . . . . . . . . 15 ⊢ (((ℤ≥‘𝑖) ∪ (1..^𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
92	90, 91	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ (((1..^𝑖) ∪ (ℤ≥‘𝑖)) ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖))
93	88, 92	syl6eq 2672	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) = ((ℤ≥‘𝑖) ∖ (1..^𝑖)))
94		difss 3737	. . . . . . . . . . . . 13 ⊢ ((ℤ≥‘𝑖) ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖)
95	93, 94	syl6eqss 3655	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖))
96		ssralv 3666	. . . . . . . . . . . 12 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖) → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
97	95, 96	syl 17	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
98		impexp 462	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)))
99		eldif 3584	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℕ ∖ (1..^𝑖)) ↔ (𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)))
100	99	imbi1i 339	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ((𝑘 ∈ ℕ ∧ ¬ 𝑘 ∈ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
101		con34b 306	. . . . . . . . . . . . . . . 16 ⊢ ((((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
102	101	imbi2i 326	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))) ↔ (𝑘 ∈ ℕ → (¬ 𝑘 ∈ (1..^𝑖) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)))
103	98, 100, 102	3bitr4i 292	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐) ↔ (𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
104	103	albii 1747	. . . . . . . . . . . . 13 ⊢ (∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
105		df-ral 2917	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ ∀𝑘(𝑘 ∈ (ℕ ∖ (1..^𝑖)) → ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
106		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑐 ∈ V
107	55	mptiniseg 5629	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ V → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐})
108	106, 107	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) = {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐}
109	108	sseq1i 3629	. . . . . . . . . . . . . 14 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ {𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖))
110		rabss 3679	. . . . . . . . . . . . . 14 ⊢ ({𝑘 ∈ ℕ ∣ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐} ⊆ (1..^𝑖) ↔ ∀𝑘 ∈ ℕ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)))
111		df-ral 2917	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ ℕ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖)) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
112	109, 110, 111	3bitri 286	. . . . . . . . . . . . 13 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) ↔ ∀𝑘(𝑘 ∈ ℕ → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → 𝑘 ∈ (1..^𝑖))))
113	104, 105, 112	3bitr4i 292	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 ↔ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖))
114		fzofi 12773	. . . . . . . . . . . . 13 ⊢ (1..^𝑖) ∈ Fin
115		ssfi 8180	. . . . . . . . . . . . 13 ⊢ (((1..^𝑖) ∈ Fin ∧ (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖)) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
116	114, 115	mpan 706	. . . . . . . . . . . 12 ⊢ ((◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ⊆ (1..^𝑖) → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
117	113, 116	sylbi 207	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ (ℕ ∖ (1..^𝑖)) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
118	97, 117	syl6 35	. . . . . . . . . 10 ⊢ (𝑖 ∈ ℕ → (∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
119	118	rexlimiv 3027	. . . . . . . . 9 ⊢ (∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → (◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
120	119	ralimi 2952	. . . . . . . 8 ⊢ (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∃𝑖 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑖) ¬ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
121	82, 120	sylbir 225	. . . . . . 7 ⊢ (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
122		iunfi 8254	. . . . . . . 8 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin ∧ ∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin) → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin)
123	122	ex 450	. . . . . . 7 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → (∀𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
124	121, 123	syl5 34	. . . . . 6 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → (¬ ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∪ 𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))(◡(𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ {𝑐}) ∈ Fin))
125	75, 124	mt3i 141	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐)
126		ssrexv 3667	. . . . 5 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 → (∃𝑐 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
127	58, 125, 126	syl2im 40	. . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐))
128		unitssre 12319	. . . . . . . . . . . 12 ⊢ (0[,]1) ⊆ ℝ
129		elmapi 7879	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0[,]1) ↑𝑚 (1...𝑁)) → 𝑐:(1...𝑁)⟶(0[,]1))
130	129, 48	eleq2s 2719	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ 𝐼 → 𝑐:(1...𝑁)⟶(0[,]1))
131	130	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ (0[,]1))
132	128, 131	sseldi 3601	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ℝ)
133		nnrp 11842	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℝ+)
134	133	rpreccld 11882	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ+)
135		eqid 2622	. . . . . . . . . . . . 13 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
136	135	rexmet 22594	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
137		blcntr 22218	. . . . . . . . . . . 12 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
138	136, 137	mp3an1 1411	. . . . . . . . . . 11 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ+) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
139	132, 134, 138	syl2an 494	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
140	139	an32s 846	. . . . . . . . 9 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
141		fveq1 6190	. . . . . . . . . 10 ⊢ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) = (𝑐‘𝑚))
142	141	eleq1d 2686	. . . . . . . . 9 ⊢ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ((((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
143	140, 142	syl5ibrcom 237	. . . . . . . 8 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
144	143	ralrimdva 2969	. . . . . . 7 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
145	144	reximdv 3016	. . . . . 6 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
146	145	ralimdva 2962	. . . . 5 ⊢ (𝑐 ∈ 𝐼 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
147	146	reximia 3009	. . . 4 ⊢ (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) = 𝑐 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
148	127, 147	syl6 35	. . 3 ⊢ (𝜑 → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
149		poimir.r	. . . . . . . 8 ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
150	51, 50	ixpconst 7918	. . . . . . . . 9 ⊢ X𝑛 ∈ (1...𝑁)(0[,]1) = ((0[,]1) ↑𝑚 (1...𝑁))
151	48, 150	eqtr4i 2647	. . . . . . . 8 ⊢ 𝐼 = X𝑛 ∈ (1...𝑁)(0[,]1)
152	149, 151	oveq12i 6662	. . . . . . 7 ⊢ (𝑅 ↾t 𝐼) = ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1))
153		fzfid 12772	. . . . . . . . 9 ⊢ (⊤ → (1...𝑁) ∈ Fin)
154		retop 22565	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
155	154	fconst6 6095	. . . . . . . . . 10 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top
156	155	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top)
157	50	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ (1...𝑁)) → (0[,]1) ∈ V)
158	153, 156, 157	ptrest 33408	. . . . . . . 8 ⊢ (⊤ → ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1)) = (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)))))
159	158	trud 1493	. . . . . . 7 ⊢ ((∏t‘((1...𝑁) × {(topGen‘ran (,))})) ↾t X𝑛 ∈ (1...𝑁)(0[,]1)) = (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))))
160		fvex 6201	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ V
161	160	fvconst2 6469	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
162	161	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝑁) → ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)) = ((topGen‘ran (,)) ↾t (0[,]1)))
163	162	mpteq2ia 4740	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
164		fconstmpt 5163	. . . . . . . . 9 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}) = (𝑛 ∈ (1...𝑁) ↦ ((topGen‘ran (,)) ↾t (0[,]1)))
165	163, 164	eqtr4i 2647	. . . . . . . 8 ⊢ (𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1))) = ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})
166	165	fveq2i 6194	. . . . . . 7 ⊢ (∏t‘(𝑛 ∈ (1...𝑁) ↦ ((((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↾t (0[,]1)))) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
167	152, 159, 166	3eqtri 2648	. . . . . 6 ⊢ (𝑅 ↾t 𝐼) = (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}))
168		fzfi 12771	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
169		dfii2 22685	. . . . . . . . 9 ⊢ II = ((topGen‘ran (,)) ↾t (0[,]1))
170		iicmp 22689	. . . . . . . . 9 ⊢ II ∈ Comp
171	169, 170	eqeltrri 2698	. . . . . . . 8 ⊢ ((topGen‘ran (,)) ↾t (0[,]1)) ∈ Comp
172	171	fconst6 6095	. . . . . . 7 ⊢ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp
173		ptcmpfi 21616	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))}):(1...𝑁)⟶Comp) → (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp)
174	168, 172, 173	mp2an 708	. . . . . 6 ⊢ (∏t‘((1...𝑁) × {((topGen‘ran (,)) ↾t (0[,]1))})) ∈ Comp
175	167, 174	eqeltri 2697	. . . . 5 ⊢ (𝑅 ↾t 𝐼) ∈ Comp
176		rehaus 22602	. . . . . . . . . . . 12 ⊢ (topGen‘ran (,)) ∈ Haus
177	176	fconst6 6095	. . . . . . . . . . 11 ⊢ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus
178		pthaus 21441	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Haus) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus)
179	168, 177, 178	mp2an 708	. . . . . . . . . 10 ⊢ (∏t‘((1...𝑁) × {(topGen‘ran (,))})) ∈ Haus
180	149, 179	eqeltri 2697	. . . . . . . . 9 ⊢ 𝑅 ∈ Haus
181		haustop 21135	. . . . . . . . 9 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Top)
182	180, 181	ax-mp 5	. . . . . . . 8 ⊢ 𝑅 ∈ Top
183		reex 10027	. . . . . . . . . 10 ⊢ ℝ ∈ V
184		mapss 7900	. . . . . . . . . 10 ⊢ ((ℝ ∈ V ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁)))
185	183, 128, 184	mp2an 708	. . . . . . . . 9 ⊢ ((0[,]1) ↑𝑚 (1...𝑁)) ⊆ (ℝ ↑𝑚 (1...𝑁))
186	48, 185	eqsstri 3635	. . . . . . . 8 ⊢ 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))
187		uniretop 22566	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
188	149, 187	ptuniconst 21401	. . . . . . . . . 10 ⊢ (((1...𝑁) ∈ Fin ∧ (topGen‘ran (,)) ∈ Top) → (ℝ ↑𝑚 (1...𝑁)) = ∪ 𝑅)
189	168, 154, 188	mp2an 708	. . . . . . . . 9 ⊢ (ℝ ↑𝑚 (1...𝑁)) = ∪ 𝑅
190	189	restuni 20966	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝐼 ⊆ (ℝ ↑𝑚 (1...𝑁))) → 𝐼 = ∪ (𝑅 ↾t 𝐼))
191	182, 186, 190	mp2an 708	. . . . . . 7 ⊢ 𝐼 = ∪ (𝑅 ↾t 𝐼)
192	191	bwth 21213	. . . . . 6 ⊢ (((𝑅 ↾t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ ¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin) → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))))
193	192	3expia 1267	. . . . 5 ⊢ (((𝑅 ↾t 𝐼) ∈ Comp ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼) → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))))
194	175, 58, 193	sylancr 695	. . . 4 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))))
195		cmptop 21198	. . . . . . . . 9 ⊢ ((𝑅 ↾t 𝐼) ∈ Comp → (𝑅 ↾t 𝐼) ∈ Top)
196	175, 195	ax-mp 5	. . . . . . . 8 ⊢ (𝑅 ↾t 𝐼) ∈ Top
197	191	islp3 20950	. . . . . . . 8 ⊢ (((𝑅 ↾t 𝐼) ∈ Top ∧ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
198	196, 197	mp3an1 1411	. . . . . . 7 ⊢ ((ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ⊆ 𝐼 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
199	58, 198	sylan 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
200		fzfid 12772	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (1...𝑁) ∈ Fin)
201	155	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ((1...𝑁) × {(topGen‘ran (,))}):(1...𝑁)⟶Top)
202		nnrecre 11057	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ)
203	202	rexrd 10089	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (1 / 𝑖) ∈ ℝ*)
204		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
205	135, 204	tgioo 22599	. . . . . . . . . . . . . . . . . 18 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
206	205	blopn 22305	. . . . . . . . . . . . . . . . 17 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
207	136, 206	mp3an1 1411	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐‘𝑚) ∈ ℝ ∧ (1 / 𝑖) ∈ ℝ*) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
208	132, 203, 207	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑚 ∈ (1...𝑁)) ∧ 𝑖 ∈ ℕ) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
209	208	an32s 846	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (topGen‘ran (,)))
210	160	fvconst2 6469	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
211	210	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑚) = (topGen‘ran (,)))
212	209, 211	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
213		noel 3919	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑚 ∈ ∅
214		difid 3948	. . . . . . . . . . . . . . . . 17 ⊢ ((1...𝑁) ∖ (1...𝑁)) = ∅
215	214	eleq2i 2693	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) ↔ 𝑚 ∈ ∅)
216	213, 215	mtbir 313	. . . . . . . . . . . . . . 15 ⊢ ¬ 𝑚 ∈ ((1...𝑁) ∖ (1...𝑁))
217	216	pm2.21i 116	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ((1...𝑁) ∖ (1...𝑁)) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
218	217	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) ∧ 𝑚 ∈ ((1...𝑁) ∖ (1...𝑁))) → ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑚))
219	200, 201, 200, 212, 218	ptopn 21386	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ (∏t‘((1...𝑁) × {(topGen‘ran (,))})))
220	219, 149	syl6eleqr 2712	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅)
221		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ↑𝑚 (1...𝑁)) ∈ V
222	48, 221	eqeltri 2697	. . . . . . . . . . . 12 ⊢ 𝐼 ∈ V
223		elrestr 16089	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Haus ∧ 𝐼 ∈ V ∧ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅) → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
224	180, 222, 223	mp3an12 1414	. . . . . . . . . . 11 ⊢ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∈ 𝑅 → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
225	220, 224	syl 17	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼))
226		difss 3737	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))
227		imassrn 5477	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ⊆ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
228	226, 227	sstri 3612	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
229	228, 58	syl5ss 3614	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼)
230		haust1 21156	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Haus → 𝑅 ∈ Fre)
231	180, 230	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 𝑅 ∈ Fre
232		restt1 21171	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Fre ∧ 𝐼 ∈ V) → (𝑅 ↾t 𝐼) ∈ Fre)
233	231, 222, 232	mp2an 708	. . . . . . . . . . . 12 ⊢ (𝑅 ↾t 𝐼) ∈ Fre
234		funmpt 5926	. . . . . . . . . . . . . 14 ⊢ Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
235		imafi 8259	. . . . . . . . . . . . . 14 ⊢ ((Fun (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ (1..^𝑖) ∈ Fin) → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin)
236	234, 114, 235	mp2an 708	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin
237		diffi 8192	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∈ Fin → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin)
238	236, 237	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin
239	191	t1ficld 21131	. . . . . . . . . . . 12 ⊢ (((𝑅 ↾t 𝐼) ∈ Fre ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ Fin) → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
240	233, 238, 239	mp3an13 1415	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ⊆ 𝐼 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
241	229, 240	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼)))
242	191	difopn 20838	. . . . . . . . . 10 ⊢ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∈ (𝑅 ↾t 𝐼) ∧ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ∈ (Clsd‘(𝑅 ↾t 𝐼))) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
243	225, 241, 242	syl2anr 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ)) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
244	243	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼))
245		eleq2 2690	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑐 ∈ 𝑣 ↔ 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))))
246		ineq1 3807	. . . . . . . . . . . 12 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) = (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
247	246	neeq1d 2853	. . . . . . . . . . 11 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅))
248	245, 247	imbi12d 334	. . . . . . . . . 10 ⊢ (𝑣 = ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → ((𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) ↔ (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)))
249	248	rspcva 3307	. . . . . . . . 9 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → (𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅))
250		ffn 6045	. . . . . . . . . . . . . . . 16 ⊢ (𝑐:(1...𝑁)⟶(0[,]1) → 𝑐 Fn (1...𝑁))
251	130, 250	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ 𝐼 → 𝑐 Fn (1...𝑁))
252	251	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 Fn (1...𝑁))
253	140	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
254	106	elixp 7915	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑐 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑐‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
255	252, 253, 254	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
256		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ 𝐼)
257	255, 256	elind 3798	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼))
258		neldifsnd 4322	. . . . . . . . . . . 12 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → ¬ 𝑐 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}))
259	257, 258	eldifd 3585	. . . . . . . . . . 11 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
260	259	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → 𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
261		simplr 792	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) → ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
262	261	anim1i 592	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
263		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
264	262, 263	anim12i 590	. . . . . . . . . . . . . . 15 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → ((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))))
265		elin 3796	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
266		andir 912	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ∨ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
267		eldif 3584	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∧ ¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})))
268		elin 3796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ↔ (𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ 𝑗 ∈ 𝐼))
269		vex 3203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑗 ∈ V
270	269	elixp 7915	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
271	270	anbi1i 731	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑗 ∈ X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ 𝑗 ∈ 𝐼) ↔ ((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼))
272	268, 271	bitri 264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ↔ ((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼))
273		ianor 509	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
274		eldif 3584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ ¬ 𝑗 ∈ {𝑐}))
275	273, 274	xchnxbir 323	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐}) ↔ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐}))
276	272, 275	anbi12i 733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∧ ¬ 𝑗 ∈ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐})))
277		andi 911	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∨ ¬ ¬ 𝑗 ∈ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})))
278	267, 276, 277	3bitri 286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})))
279		eldif 3584	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐}) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
280	278, 279	anbi12i 733	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∨ (((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐})) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
281		pm3.24 926	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐})
282		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) → ¬ ¬ 𝑗 ∈ {𝑐})
283		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}) → ¬ 𝑗 ∈ {𝑐})
284	282, 283	anim12ci 591	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) → (¬ 𝑗 ∈ {𝑐} ∧ ¬ ¬ 𝑗 ∈ {𝑐}))
285	281, 284	mto 188	. . . . . . . . . . . . . . . . . 18 ⊢ ¬ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))
286	285	biorfi 422	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ↔ (((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})) ∨ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ ¬ 𝑗 ∈ {𝑐}) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐}))))
287	266, 280, 286	3bitr4i 292	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∧ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
288	265, 287	bitri 264	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ↔ ((((𝑗 Fn (1...𝑁) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ∧ 𝑗 ∈ 𝐼) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ {𝑐})))
289		ancom 466	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ↔ (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))))
290		anass 681	. . . . . . . . . . . . . . . 16 ⊢ (((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ (¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))))
291	289, 290	bitr4i 267	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) ↔ ((∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))))
292	264, 288, 291	3imtr4i 281	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
293		ancom 466	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
294		eldif 3584	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ↔ (𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
295	293, 294	bitr4i 267	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ↔ 𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))))
296		imadmrn 5476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) = ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
297	66, 55	dmmpti 6023	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) = ℕ
298	297	imaeq2i 5464	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ dom (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ℕ)
299	296, 298	eqtr3i 2646	. . . . . . . . . . . . . . . . . . . 20 ⊢ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) = ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ℕ)
300	299	difeq1i 3724	. . . . . . . . . . . . . . . . . . 19 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) = (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)))
301		imadifss 33384	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ ℕ) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
302	300, 301	eqsstri 3635	. . . . . . . . . . . . . . . . . 18 ⊢ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖)))
303		imass2 5501	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℕ ∖ (1..^𝑖)) ⊆ (ℤ≥‘𝑖) → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)))
304	95, 303	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)))
305		df-ima 5127	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)) = ran ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖))
306		uznnssnn 11735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ ℕ → (ℤ≥‘𝑖) ⊆ ℕ)
307	306	resmptd 5452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖)) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
308	307	rneqd 5353	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ ℕ → ran ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ↾ (ℤ≥‘𝑖)) = ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
309	305, 308	syl5eq 2668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℤ≥‘𝑖)) = ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
310	304, 309	sseqtrd 3641	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ → ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (ℕ ∖ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
311	302, 310	syl5ss 3614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℕ → (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) ⊆ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))))
312	311	sseld 3602	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℕ → (𝑗 ∈ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))))
313	295, 312	syl5bi 232	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℕ → ((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) → 𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))))
314	313	anim1d 588	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ℕ → (((¬ 𝑗 ∈ ((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∧ 𝑗 ∈ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))) → (𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
315	292, 314	syl5 34	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℕ → (𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → (𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
316	315	eximdv 1846	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ ℕ → (∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) → ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))))
317		n0 3931	. . . . . . . . . . . 12 ⊢ ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ ↔ ∃𝑗 𝑗 ∈ (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})))
318	66	rgenw 2924	. . . . . . . . . . . . . 14 ⊢ ∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ V
319		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) = (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))
320		fveq1 6190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) → (𝑗‘𝑚) = (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚))
321	320	eleq1d 2686	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) → ((𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ (((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
322	321	ralbidv 2986	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) → (∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
323	319, 322	rexrnmpt 6369	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑖)((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})) ∈ V → (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
324	318, 323	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
325		df-rex 2918	. . . . . . . . . . . . 13 ⊢ (∃𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
326	324, 325	bitr3i 266	. . . . . . . . . . . 12 ⊢ (∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ↔ ∃𝑗(𝑗 ∈ ran (𝑘 ∈ (ℤ≥‘𝑖) ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∧ ∀𝑚 ∈ (1...𝑁)(𝑗‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
327	316, 317, 326	3imtr4g 285	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
328	327	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅ → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
329	260, 328	embantd 59	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((𝑐 ∈ ((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) → (((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
330	249, 329	syl5 34	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → ((((X𝑚 ∈ (1...𝑁)((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) ∩ 𝐼) ∖ (((𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) “ (1..^𝑖)) ∖ {𝑐})) ∈ (𝑅 ↾t 𝐼) ∧ ∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅)) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
331	244, 330	mpand 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝐼) ∧ 𝑖 ∈ ℕ) → (∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
332	331	ralrimdva 2969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → (𝑣 ∩ (ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∖ {𝑐})) ≠ ∅) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
333	199, 332	sylbid 230	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) → ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
334	333	reximdva 3017	. . . 4 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 𝑐 ∈ ((limPt‘(𝑅 ↾t 𝐼))‘ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘})))) → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
335	194, 334	syld 47	. . 3 ⊢ (𝜑 → (¬ ran (𝑘 ∈ ℕ ↦ ((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))) ∈ Fin → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖))))
336	148, 335	pm2.61d 170	. 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)))
337		poimir.0	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
338		poimir.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))
339		poimirlem30.x	. . . 4 ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘𝑓 + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘𝑓 / ((1...𝑁) × {𝑘})))‘𝑛)
340		poimirlem30.4	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)
341	337, 48, 149, 338, 339, 34, 39, 340	poimirlem29 33438	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
342	341	reximdv 3016	. 2 ⊢ (𝜑 → (∃𝑐 ∈ 𝐼 ∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘𝑓 / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝑐‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
343	336, 342	mpd 15	1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ⊤wtru 1484  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ωcom 7065  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Xcixp 7908   ≈ cen 7952  Fincfn 7955  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℕ₀cn0 11292  ℤ_≥cuz 11687  ℝ⁺crp 11832  (,)cioo 12175  [,]cicc 12178  ...cfz 12326  ..^cfzo 12465  abscabs 13974   ↾_t crest 16081  topGenctg 16098  ∏_tcpt 16099  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  Topctop 20698  Clsdccld 20820  limPtclp 20938   Cn ccn 21028  Frect1 21111  Hauscha 21112  Compccmp 21189  IIcii 22678

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-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-iin 4523  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-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-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-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-rest 16083  df-topgen 16104  df-pt 16105  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-cld 20823  df-ntr 20824  df-cls 20825  df-lp 20940  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-hmph 21559  df-ii 22680

This theorem is referenced by:  poimirlem32  33441