dvnprodlem1 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem1 40161

Description: 𝐷 is bijective. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnprodlem1.c	⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
dvnprodlem1.j	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
dvnprodlem1.d	⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
dvnprodlem1.t	⊢ (𝜑 → 𝑇 ∈ Fin)
dvnprodlem1.z	⊢ (𝜑 → 𝑍 ∈ 𝑇)
dvnprodlem1.zr	⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
dvnprodlem1.rzt	⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)

**Assertion**
Ref	Expression
dvnprodlem1	⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))

Distinct variable groups: 𝐶,𝑐,𝑘 𝐷,𝑐,𝑡 𝐽,𝑐,𝑘,𝑛,𝑡 𝑅,𝑐,𝑘,𝑛,𝑡 𝑅,𝑠,𝑐,𝑛,𝑡 𝑡,𝑇,𝑠 𝑍,𝑐,𝑘,𝑛,𝑡 𝑍,𝑠 𝜑,𝑐,𝑛,𝑡,𝑘 𝜑,𝑠 Allowed substitution hints: 𝐶(𝑡,𝑛,𝑠) 𝐷(𝑘,𝑛,𝑠) 𝑇(𝑘,𝑛,𝑐) 𝐽(𝑠)

Proof of Theorem dvnprodlem1 Dummy variables 𝑑 𝑒 𝑚 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
2		0zd 11389	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ∈ ℤ)
3		dvnprodlem1.j	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
4	3	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ ℤ)
5	4	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℤ)
6		fzssz 12343	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℤ
7	6	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℤ)
8		dvnprodlem1.c	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
9	8	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})))
10		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})))
11		rabeq 3192	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
12	10, 11	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
13		sumeq1 14419	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
14	13	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛))
15	14	rabbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
16	12, 15	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
17	16	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
18	17	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 = (𝑅 ∪ {𝑍})) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
19		dvnprodlem1.rzt	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
20		dvnprodlem1.t	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑇 ∈ Fin)
21		ssexg 4804	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ∪ {𝑍}) ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ V)
22	19, 20, 21	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
23		elpwg 4166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
24	22, 23	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
25	19, 24	mpbird 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
26		nn0ex 11298	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ₀ ∈ V
27	26	mptex 6486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V)
29	9, 18, 25, 28	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
30		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
31	30	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
32		rabeq 3192	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})
34		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
35	34	rabbidv 3189	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
36	33, 35	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
37	36	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑛 = 𝐽) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
38		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∈ V
39	38	rabex 4813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V)
41	29, 37, 3, 40	fvmptd 6288	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
42		ssrab2 3687	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍}))
43	42	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
44	41, 43	eqsstrd 3639	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
45	44	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
46		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
47	45, 46	sseldd 3604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
48		elmapi 7879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
49	47, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
50		dvnprodlem1.z	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
51		snidg 4206	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍})
52	50, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
53		elun2 3781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍}))
55	54	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
56	49, 55	ffvelrnd 6360	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽))
57	7, 56	sseldd 3604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℤ)
58	5, 57	zsubcld 11487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
59	2, 5, 58	3jca 1242	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ))
60		elfzle2 12345	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽)
61	56, 60	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ≤ 𝐽)
62	5	zred 11482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℝ)
63	57	zred 11482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℝ)
64	62, 63	subge0d 10617	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽))
65	61, 64	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍)))
66		elfzle1 12344	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍))
67	56, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝑐‘𝑍))
68	62, 63	subge02d 10619	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))
69	67, 68	mpbid 222	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)
70	59, 65, 69	jca32 558	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
71		elfz2 12333	. . . . . . . . . . 11 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
72	70, 71	sylibr 224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
73		elmapfn 7880	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → 𝑐 Fn (𝑅 ∪ {𝑍}))
74	47, 73	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
75		ssun1 3776	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 ⊆ (𝑅 ∪ {𝑍})
76	75	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
77		fnssres 6004	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑅 ⊆ (𝑅 ∪ {𝑍})) → (𝑐 ↾ 𝑅) Fn 𝑅)
78	74, 76, 77	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) Fn 𝑅)
79		nfv 1843	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡𝜑
80		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡𝑐
81		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑡𝒫 𝑇
82		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑡ℕ₀
83		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ Ⅎ𝑡𝑠
84	83	nfsum1 14420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡)
85		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑡𝑛
86	84, 85	nfeq 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛
87		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑡((0...𝑛) ↑_𝑚 𝑠)
88	86, 87	nfrab 3123	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑡{𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}
89	82, 88	nfmpt 4746	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
90	81, 89	nfmpt 4746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑡(𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
91	8, 90	nfcxfr 2762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡𝐶
92		nfcv 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡(𝑅 ∪ {𝑍})
93	91, 92	nffv 6198	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡(𝐶‘(𝑅 ∪ {𝑍}))
94		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡𝐽
95	93, 94	nffv 6198	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
96	80, 95	nfel 2777	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
97	79, 96	nfan 1828	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
98		fvres 6207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
99	98	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
100	2	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ∈ ℤ)
101	58	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
102	6	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℤ)
103	49	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
104	76	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
105	103, 104	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽))
106	102, 105	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℤ)
107	100, 101, 106	3jca 1242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ))
108		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑐‘𝑡) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑡))
109	105, 108	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ≤ (𝑐‘𝑡))
110	19	unssad 3790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑅 ⊆ 𝑇)
111		ssfi 8180	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin)
112	20, 110, 111	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ∈ Fin)
113	112	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑅 ∈ Fin)
114		zssre 11384	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℤ ⊆ ℝ
115	6, 114	sstri 3612	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0...𝐽) ⊆ ℝ
116	115	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (0...𝐽) ⊆ ℝ)
117	49	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
118	76	sselda 3603	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ (𝑅 ∪ {𝑍}))
119	117, 118	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ (0...𝐽))
120	116, 119	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ)
121	120	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ)
122		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑐‘𝑟) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑟))
123	119, 122	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟))
124	123	adantlr 751	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟))
125		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑟 = 𝑡 → (𝑐‘𝑟) = (𝑐‘𝑡))
126		simpr 477	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅)
127	113, 121, 124, 125, 126	fsumge1 14529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
128	112	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
129	120	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℂ)
130	128, 129	fsumcl 14464	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) ∈ ℂ)
131	63	recnd 10068	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ)
132	130, 131	pncand 10393	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
133		nfv 1843	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑟(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
134		nfcv 2764	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Ⅎ𝑟(𝑐‘𝑍)
135	50	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
136		dvnprodlem1.zr	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
137	136	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
138		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑟 = 𝑍 → (𝑐‘𝑟) = (𝑐‘𝑍))
139	133, 134, 128, 135, 137, 129, 138, 131	fsumsplitsn 14474	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)))
140	139	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟))
141	125	cbvsumv 14426	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)
142	141	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
143	41	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
144	46, 143	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
145		rabid 3116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
146	144, 145	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
147	146	simprd 479	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)
148	142, 147	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = 𝐽)
149	140, 148	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = 𝐽)
150	149	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = (𝐽 − (𝑐‘𝑍)))
151	132, 150	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
152	151	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
153	127, 152	breqtrd 4679	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))
154	107, 109, 153	jca32 558	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))))
155		elfz2 12333	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))) ↔ ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍)))))
156	154, 155	sylibr 224	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
157	99, 156	eqeltrd 2701	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
158	157	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
159	97, 158	ralrimi 2957	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))
160	78, 159	jca 554	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
161		ffnfv 6388	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))) ↔ ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))))
162	160, 161	sylibr 224	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))))
163		ovexd 6680	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...(𝐽 − (𝑐‘𝑍))) ∈ V)
164	20, 110	ssexd 4805	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ V)
165	164	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ V)
166	163, 165	elmapd 7871	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ↔ (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍)))))
167	162, 166	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅))
168	98	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)))
169	97, 168	ralrimi 2957	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
170	169	sumeq2d 14432	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
171	125	cbvsumv 14426	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)
172	171	eqcomi 2631	. . . . . . . . . . . . . . 15 ⊢ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)
173	172	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟))
174	151	idi 2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍)))
175	170, 173, 174	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))
176	167, 175	jca 554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
177		eqidd 2623	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → 𝑅 = 𝑅)
178		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → 𝑒 = (𝑐 ↾ 𝑅))
179	178	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → (𝑒‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
180	177, 179	sumeq12rdv 14438	. . . . . . . . . . . . . 14 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡))
181	180	eqeq1d 2624	. . . . . . . . . . . . 13 ⊢ (𝑒 = (𝑐 ↾ 𝑅) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)) ↔ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
182	181	elrab 3363	. . . . . . . . . . . 12 ⊢ ((𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ↔ ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))))
183	176, 182	sylibr 224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
184		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑅))
185		rabeq 3192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
186	184, 185	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
187		sumeq1 14419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
188	187	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛))
189	188	rabbidv 3189	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
190	186, 189	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
191	190	mpteq2dv 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = 𝑅 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
192	191	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 = 𝑅) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
193		elpwg 4166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
194	164, 193	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
195	110, 194	mpbird 247	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇)
196	26	mptex 6486	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V
197	196	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V)
198	9, 192, 195, 197	fvmptd 6288	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
199	198	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
200		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚))
201	200	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑚) ↑_𝑚 𝑅))
202		rabeq 3192	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑚) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
203	201, 202	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
204		eqeq2 2633	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚))
205	204	rabbidv 3189	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
206	203, 205	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
207	206	cbvmptv 4750	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})
208	207	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}))
209	199, 208	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑚 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}))
210		fveq1 6190	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡))
211	210	sumeq2ad 14434	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑒‘𝑡))
212	211	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚))
213	212	cbvrabv 3199	. . . . . . . . . . . . . . . 16 ⊢ {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}
214	213	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
215		oveq2 6658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (0...𝑚) = (0...(𝐽 − (𝑐‘𝑍))))
216	215	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → ((0...𝑚) ↑_𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅))
217		rabeq 3192	. . . . . . . . . . . . . . . 16 ⊢ (((0...𝑚) ↑_𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) → {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
218	216, 217	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚})
219		eqeq2 2633	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))))
220	219	rabbidv 3189	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
221	214, 218, 220	3eqtrd 2660	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
222	221	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑚 = (𝐽 − (𝑐‘𝑍))) → {𝑐 ∈ ((0...𝑚) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
223	58, 65	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍))))
224		elnn0z 11390	. . . . . . . . . . . . . 14 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ ℕ₀ ↔ ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍))))
225	223, 224	sylibr 224	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
226		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∈ V
227	226	rabex 4813	. . . . . . . . . . . . . 14 ⊢ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V
228	227	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V)
229	209, 222, 225, 228	fvmptd 6288	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))) = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))})
230	229	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
231	183, 230	eleqtrd 2703	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
232	72, 231	jca 554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))
233	1, 232	jca 554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))))
234		ovexd 6680	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ V)
235		vex 3203	. . . . . . . . . . 11 ⊢ 𝑐 ∈ V
236	235	resex 5443	. . . . . . . . . 10 ⊢ (𝑐 ↾ 𝑅) ∈ V
237	236	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ V)
238		opeq12 4404	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ⟨𝑘, 𝑑⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
239	238	eqeq2d 2632	. . . . . . . . . . 11 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ↔ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
240		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)))
241	240	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)))
242		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 𝑑 = (𝑐 ↾ 𝑅))
243		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
244	243	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))
245	242, 244	eleq12d 2695	. . . . . . . . . . . 12 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑑 ∈ ((𝐶‘𝑅)‘𝑘) ↔ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))
246	241, 245	anbi12d 747	. . . . . . . . . . 11 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))))
247	239, 246	anbi12d 747	. . . . . . . . . 10 ⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))) ↔ (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))))
248	247	spc2egv 3295	. . . . . . . . 9 ⊢ (((𝐽 − (𝑐‘𝑍)) ∈ V ∧ (𝑐 ↾ 𝑅) ∈ V) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))))
249	234, 237, 248	syl2anc 693	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))))
250	233, 249	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))
251		eliunxp 5259	. . . . . . 7 ⊢ (⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘∃𝑑(⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨𝑘, 𝑑⟩ ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))
252	250, 251	sylibr 224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
253		dvnprodlem1.d	. . . . . 6 ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
254	252, 253	fmptd 6385	. . . . 5 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
255	95	nfcri 2758	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)
256	96, 255	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
257	79, 256	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
258		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝐷‘𝑐) = (𝐷‘𝑒)
259	257, 258	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑡((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒))
260	99	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
261	260	adantlrr 757	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
262	261	adantlr 751	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
263	253	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
264		opex 4932	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V
265	264	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V)
266	263, 265	fvmpt2d 6293	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
267	266	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
268	267	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡))
269		ovex 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 − (𝑐‘𝑍)) ∈ V
270	269, 236	op2nd 7177	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝑐 ↾ 𝑅)
271	270	fveq1i 6192	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)
272	271	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2^nd ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
273	268, 272	eqtr2d 2657	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
274	273	adantrr 753	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
275	274	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2^nd ‘(𝐷‘𝑐))‘𝑡))
276		simpr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = (𝐷‘𝑒))
277	253	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
278		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑍) = (𝑒‘𝑍))
279	278	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝑒‘𝑍)))
280		reseq1 5390	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝑐 ↾ 𝑅) = (𝑒 ↾ 𝑅))
281	279, 280	opeq12d 4410	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑒 → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
282	281	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑐 = 𝑒) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
283		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
284		opex 4932	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩ ∈ V
285	284	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩ ∈ V)
286	277, 282, 283, 285	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑒) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
287	286	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑒) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
288	276, 287	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩)
289	288	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
290	289	adantlrl 756	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
291	290	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘(𝐷‘𝑐)) = (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
292		ovex 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 − (𝑒‘𝑍)) ∈ V
293		vex 3203	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑒 ∈ V
294	293	resex 5443	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ↾ 𝑅) ∈ V
295	292, 294	op2nd 7177	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝑒 ↾ 𝑅)
296	295	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝑒 ↾ 𝑅))
297	291, 296	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2^nd ‘(𝐷‘𝑐)) = (𝑒 ↾ 𝑅))
298	297	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = ((𝑒 ↾ 𝑅)‘𝑡))
299		fvres 6207	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑅 → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡))
300	299	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡))
301	298, 300	eqtrd 2656	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘(𝐷‘𝑐))‘𝑡) = (𝑒‘𝑡))
302	262, 275, 301	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
303	302	adantlr 751	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
304		simpl 473	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
305		elunnel1 3754	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 ∈ {𝑍})
306		elsni 4194	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ {𝑍} → 𝑡 = 𝑍)
307	305, 306	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
308	307	adantll 750	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
309		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍)
310	309	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑐‘𝑍))
311	3	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ ℂ)
312	311	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
313	312, 131	nncand 10397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍))
314	313	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
315	314	adantrr 753	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
316	315	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
317	266	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
318	269, 236	op1st 7176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝐽 − (𝑐‘𝑍))
319	318	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩) = (𝐽 − (𝑐‘𝑍)))
320	317, 319	eqtr2d 2657	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) = (1^st ‘(𝐷‘𝑐)))
321	320	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
322	321	adantrr 753	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
323	322	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1^st ‘(𝐷‘𝑐))))
324		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐷‘𝑐) = (𝐷‘𝑒) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘(𝐷‘𝑒)))
325	324	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑐)) = (1^st ‘(𝐷‘𝑒)))
326	286	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1^st ‘(𝐷‘𝑒)) = (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
327	326	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑒)) = (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩))
328	292, 294	op1st 7176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝐽 − (𝑒‘𝑍))
329	328	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘⟨(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)⟩) = (𝐽 − (𝑒‘𝑍)))
330	325, 327, 329	3eqtrd 2660	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1^st ‘(𝐷‘𝑐)) = (𝐽 − (𝑒‘𝑍)))
331	330	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝐽 − (𝐽 − (𝑒‘𝑍))))
332	311	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
333		zsscn 11385	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℤ ⊆ ℂ
334	6, 333	sstri 3612	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...𝐽) ⊆ ℂ
335	334	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
336		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑐 = 𝑒 → (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↔ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))
337	336	anbi2d 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ↔ (𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))))
338		feq1 6026	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑐 = 𝑒 → (𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽) ↔ 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)))
339	337, 338	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = 𝑒 → (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) ↔ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))))
340	339, 49	chvarv 2263	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))
341	54	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
342	340, 341	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ (0...𝐽))
343	335, 342	sseldd 3604	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ ℂ)
344	332, 343	nncand 10397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍))
345	344	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍))
346		eqidd 2623	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑒‘𝑍) = (𝑒‘𝑍))
347	331, 345, 346	3eqtrd 2660	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝑒‘𝑍))
348	347	adantlrl 756	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1^st ‘(𝐷‘𝑐))) = (𝑒‘𝑍))
349	316, 323, 348	3eqtrd 2660	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝑒‘𝑍))
350	349	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑍) = (𝑒‘𝑍))
351		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → (𝑒‘𝑡) = (𝑒‘𝑍))
352	351	eqcomd 2628	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (𝑒‘𝑍) = (𝑒‘𝑡))
353	352	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑒‘𝑍) = (𝑒‘𝑡))
354	310, 350, 353	3eqtrd 2660	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡))
355	354	adantlr 751	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡))
356	304, 308, 355	syl2anc 693	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡))
357	303, 356	pm2.61dan 832	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) = (𝑒‘𝑡))
358	357	ex 450	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → (𝑐‘𝑡) = (𝑒‘𝑡)))
359	259, 358	ralrimi 2957	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))
360	74	adantrr 753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑐 Fn (𝑅 ∪ {𝑍}))
361	360	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 Fn (𝑅 ∪ {𝑍}))
362		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽) → 𝑒 Fn (𝑅 ∪ {𝑍}))
363	340, 362	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
364	363	adantrl 752	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑒 Fn (𝑅 ∪ {𝑍}))
365	364	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑒 Fn (𝑅 ∪ {𝑍}))
366		eqfnfv 6311	. . . . . . . . 9 ⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑒 Fn (𝑅 ∪ {𝑍})) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)))
367	361, 365, 366	syl2anc 693	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)))
368	359, 367	mpbird 247	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 = 𝑒)
369	368	ex 450	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒))
370	369	ralrimivva 2971	. . . . 5 ⊢ (𝜑 → ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒))
371	254, 370	jca 554	. . . 4 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)))
372		dff13 6512	. . . 4 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)))
373	371, 372	sylibr 224	. . 3 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
374		eliun 4524	. . . . . . . . . . 11 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
375	374	biimpi 206	. . . . . . . . . 10 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
376	375	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
377		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝜑
378		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑝
379		nfiu1 4550	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
380	378, 379	nfel 2777	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
381	377, 380	nfan 1828	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
382		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}
383		nfv 1843	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑘 ∈ (0...𝐽)
384		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡𝑝
385		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡{𝑘}
386		nfcv 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑡𝑅
387	91, 386	nffv 6198	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡(𝐶‘𝑅)
388		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑡𝑘
389	387, 388	nffv 6198	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑡((𝐶‘𝑅)‘𝑘)
390	385, 389	nfxp 5142	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑡({𝑘} × ((𝐶‘𝑅)‘𝑘))
391	384, 390	nfel 2777	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))
392	79, 383, 391	nf3an 1831	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
393		0zd 11389	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ∈ ℤ)
394	4	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
395	394	3ad2antl1 1223	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ)
396		iftrue 4092	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
397	396	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
398		fzssz 12343	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0...𝑘) ⊆ ℤ
399	398	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℤ)
400		simp1 1061	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
401		simp2 1062	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
402		xp2nd 7199	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
403	402	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
404	198	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
405		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
406	405	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑘) ↑_𝑚 𝑅))
407		rabeq 3192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑘) ↑_𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
408	406, 407	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
409		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘))
410	409	rabbidv 3189	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
411	408, 410	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
412	411	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
413		elfznn0 12433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ₀)
414	413	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ₀)
415		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((0...𝑘) ↑_𝑚 𝑅) ∈ V
416	415	rabex 4813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V
417	416	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V)
418	404, 412, 414, 417	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
419	418	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
420	403, 419	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
421		elrabi 3359	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
422	421	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ (2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
423	400, 401, 420, 422	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅))
424		elmapi 7879	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
425	423, 424	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
426	425	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (2^nd ‘𝑝):𝑅⟶(0...𝑘))
427	426	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘))
428	399, 427	sseldd 3604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
429	397, 428	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
430		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})))
431	307	adantll 750	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍)
432		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍)
433	136	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑍 ∈ 𝑅)
434	432, 433	eqneltrd 2720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅)
435	434	iffalsed 4097	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
436	435	3ad2antl1 1223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
437	4	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ)
438	437	3ad2antl1 1223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ)
439		xp1st 7198	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ {𝑘})
440		elsni 4194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((1^st ‘𝑝) ∈ {𝑘} → (1^st ‘𝑝) = 𝑘)
441	439, 440	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) = 𝑘)
442	441	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
443	6	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
444	443	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℤ)
445	442, 444	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
446	445	3adant1 1079	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
447	446	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (1^st ‘𝑝) ∈ ℤ)
448	438, 447	zsubcld 11487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1^st ‘𝑝)) ∈ ℤ)
449	436, 448	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
450	449	adantlr 751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
451	430, 431, 450	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
452	429, 451	pm2.61dan 832	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ)
453	393, 395, 452	3jca 1242	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ))
454	425	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘))
455		elfzle1 12344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → 0 ≤ ((2^nd ‘𝑝)‘𝑡))
456	454, 455	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ ((2^nd ‘𝑝)‘𝑡))
457	396	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 ∈ 𝑅 → ((2^nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
458	457	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
459	456, 458	breqtrd 4679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
460	459	adantlr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
461		simpll 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))))
462		elfzle2 12345	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽)
463		elfzel2 12340	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
464	463	zred 11482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
465	115	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
466	464, 465	subge0d 10617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽))
467	462, 466	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘))
468	467	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘))
469	468	3ad2antl2 1224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘))
470	400, 434	sylan 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅)
471	470	iffalsed 4097	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
472	442	3adant1 1079	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
473	472	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
474	473	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
475	471, 474	eqtr2d 2657	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
476	469, 475	breqtrd 4679	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
477	461, 431, 476	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
478	460, 477	pm2.61dan 832	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
479		simpl2 1065	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑘 ∈ (0...𝐽))
480	398	sseli 3599	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
481	480	zred 11482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ∈ ℝ)
482	481	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ∈ ℝ)
483	465	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
484	464	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
485		elfzle2 12345	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝑘)
486	485	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝑘)
487	462	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽)
488	482, 483, 484, 486, 487	letrd 10194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
489	454, 479, 488	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
490	489	adantlr 751	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ≤ 𝐽)
491	397, 490	eqbrtrd 4675	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
492	475	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (𝐽 − 𝑘))
493	414	nn0ge0d 11354	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
494	464	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
495	465	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
496	494, 495	subge02d 10619	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽))
497	493, 496	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ≤ 𝐽)
498	497	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽)
499	498	3adantl3 1219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽)
500	492, 499	eqbrtrd 4675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
501	461, 431, 500	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
502	491, 501	pm2.61dan 832	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)
503	453, 478, 502	jca32 558	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ) ∧ (0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)))
504		elfz2 12333	. . . . . . . . . . . . . . . . 17 ⊢ (if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℤ) ∧ (0 ≤ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ≤ 𝐽)))
505	503, 504	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ (0...𝐽))
506		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
507	392, 505, 506	fmptdf 6387	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽))
508		ovexd 6680	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ∈ V)
509	400, 22	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑅 ∪ {𝑍}) ∈ V)
510	508, 509	elmapd 7871	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ↔ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽)))
511	507, 510	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
512		eqidd 2623	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
513		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑡 → (𝑟 ∈ 𝑅 ↔ 𝑡 ∈ 𝑅))
514		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝑡 → ((2^nd ‘𝑝)‘𝑟) = ((2^nd ‘𝑝)‘𝑡))
515	513, 514	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝑡 → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
516	515	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑟 = 𝑡) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
517		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
518	512, 516, 517, 452	fvmptd 6288	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
519	518	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
520	392, 519	ralrimi 2957	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
521	520	sumeq2d 14432	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))
522		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))
523	400, 112	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
524	400, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇)
525	400, 136	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅)
526	396	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
527	454, 480	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℤ)
528	527	zcnd 11483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ ℂ)
529	526, 528	eqeltrd 2701	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) ∈ ℂ)
530		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑅 ↔ 𝑍 ∈ 𝑅))
531		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑍 → ((2^nd ‘𝑝)‘𝑡) = ((2^nd ‘𝑝)‘𝑍))
532	530, 531	ifbieq1d 4109	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))))
533	136	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅)
534	533	iffalsed 4097	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
535	534	3adant2 1080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
536	4	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℤ)
537	536, 446	zsubcld 11487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ ℤ)
538	537	zcnd 11483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ ℂ)
539	535, 538	eqeltrd 2701	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) ∈ ℂ)
540	392, 522, 523, 524, 525, 529, 532, 539	fsumsplitsn 14474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))))
541	396	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡)))
542	392, 541	ralrimi 2957	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = ((2^nd ‘𝑝)‘𝑡))
543	542	sumeq2d 14432	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡))
544		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = (2^nd ‘𝑝) → 𝑅 = 𝑅)
545		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑐 = (2^nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → 𝑐 = (2^nd ‘𝑝))
546	545	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑐 = (2^nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((2^nd ‘𝑝)‘𝑡))
547	544, 546	sumeq12rdv 14438	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = (2^nd ‘𝑝) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡))
548	547	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (2^nd ‘𝑝) → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘 ↔ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
549	548	elrab 3363	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2^nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ↔ ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
550	420, 549	sylib 208	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((2^nd ‘𝑝) ∈ ((0...𝑘) ↑_𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘))
551	550	simprd 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 ((2^nd ‘𝑝)‘𝑡) = 𝑘)
552	543, 551	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = 𝑘)
553	525	iffalsed 4097	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
554	553, 473	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝))) = (𝐽 − 𝑘))
555	552, 554	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))) = (𝑘 + (𝐽 − 𝑘)))
556	334	sseli 3599	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℂ)
557	556	3ad2ant2 1083	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℂ)
558	400, 311	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℂ)
559	557, 558	pncan3d 10395	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 + (𝐽 − 𝑘)) = 𝐽)
560	555, 559	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑍), (𝐽 − (1^st ‘𝑝)))) = 𝐽)
561	521, 540, 560	3eqtrd 2660	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽)
562	511, 561	jca 554	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
563		eleq1 2689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑟 → (𝑡 ∈ 𝑅 ↔ 𝑟 ∈ 𝑅))
564		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑟 → ((2^nd ‘𝑝)‘𝑡) = ((2^nd ‘𝑝)‘𝑟))
565	563, 564	ifbieq1d 4109	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑟 → if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))) = if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))
566	565	cbvmptv 4750	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))
567	566	eqeq2i 2634	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↔ 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
568	567	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
569		fveq1 6190	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) → (𝑐‘𝑡) = ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
570	569	sumeq2ad 14434	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
571	568, 570	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡))
572	571	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
573	572	elrab 3363	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))))‘𝑡) = 𝐽))
574	562, 573	sylibr 224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
575	574	3exp 1264	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})))
576	575	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})))
577	381, 382, 576	rexlimd 3026	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}))
578	376, 577	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
579	41	eqcomd 2628	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
580	579	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
581	578, 580	eleqtrd 2703	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
582	253	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩))
583		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
584	566	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
585	583, 584	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝)))))
586		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → 𝑟 = 𝑍)
587	136	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑍 ∈ 𝑅)
588	586, 587	eqneltrd 2720	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑟 ∈ 𝑅)
589	588	iffalsed 4097	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
590	589	adantlr 751	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑟), (𝐽 − (1^st ‘𝑝))) = (𝐽 − (1^st ‘𝑝)))
591	54	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
592		ovexd 6680	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (1^st ‘𝑝)) ∈ V)
593	585, 590, 591, 592	fvmptd 6288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐‘𝑍) = (𝐽 − (1^st ‘𝑝)))
594	593	oveq2d 6666	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1^st ‘𝑝))))
595	594	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1^st ‘𝑝))))
596	311	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝐽 ∈ ℂ)
597		nfv 1843	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(1^st ‘𝑝) ∈ (0...𝐽)
598		simpl 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
599		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → (1^st ‘𝑝) = 𝑘)
600		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → 𝑘 ∈ (0...𝐽))
601	599, 600	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (0...𝐽) ∧ (1^st ‘𝑝) = 𝑘) → (1^st ‘𝑝) ∈ (0...𝐽))
602	598, 442, 601	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
603	602	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
604	603	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽))))
605	380, 597, 604	rexlimd 3026	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
606	375, 605	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽))
607	6	sseli 3599	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℤ)
608	606, 607	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ ℤ)
609	608	zcnd 11483	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ ℂ)
610	609	ad2antlr 763	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (1^st ‘𝑝) ∈ ℂ)
611	596, 610	nncand 10397	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝐽 − (1^st ‘𝑝))) = (1^st ‘𝑝))
612	595, 611	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (1^st ‘𝑝))
613		reseq1 5390	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅))
614	613	adantl 482	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅))
615	75	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → 𝑅 ⊆ (𝑅 ∪ {𝑍}))
616	615	resmptd 5452	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ↾ 𝑅) = (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))
617		nfv 1843	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝)
618	396	mpteq2ia 4740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡))
619	618	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡)))
620	425	feqmptd 6249	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) = (𝑡 ∈ 𝑅 ↦ ((2^nd ‘𝑝)‘𝑡)))
621	619, 620	eqtr4d 2659	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
622	621	3exp 1264	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))))
623	622	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))))
624	381, 617, 623	rexlimd 3026	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝)))
625	376, 624	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
626	625	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) = (2^nd ‘𝑝))
627	614, 616, 626	3eqtrd 2660	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → (𝑐 ↾ 𝑅) = (2^nd ‘𝑝))
628	612, 627	opeq12d 4410	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
629		opex 4932	. . . . . . . . . 10 ⊢ ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∈ V
630	629	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ ∈ V)
631	582, 628, 581, 630	fvmptd 6288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))) = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
632		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑘⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝
633		1st2nd2 7205	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 𝑝 = ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩)
634	633	eqcomd 2628	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)
635	634	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝))
636	635	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)))
637	381, 632, 636	rexlimd 3026	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝))
638	376, 637	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ⟨(1^st ‘𝑝), (2^nd ‘𝑝)⟩ = 𝑝)
639	631, 638	eqtr2d 2657	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))))
640		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝐷‘𝑐) = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝))))))
641	640	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) → (𝑝 = (𝐷‘𝑐) ↔ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))))
642	641	rspcev 3309	. . . . . . 7 ⊢ (((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2^nd ‘𝑝)‘𝑡), (𝐽 − (1^st ‘𝑝)))))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
643	581, 639, 642	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
644	643	ralrimiva 2966	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))
645	254, 644	jca 554	. . . 4 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)))
646		dffo3 6374	. . . 4 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)))
647	645, 646	sylibr 224	. . 3 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
648	373, 647	jca 554	. 2 ⊢ (𝜑 → (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))))
649		df-f1o 5895	. 2 ⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))))
650	648, 649	sylibr 224	1 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∪ cun 3572 ⊆ wss 3574 ifcif 4086 𝒫 cpw 4158 {csn 4177 ⟨cop 4183 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 –1-1→wf1 5885 –onto→wfo 5886 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ↑_𝑚 cmap 7857 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 ≤ cle 10075 − cmin 10266 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 Σcsu 14416

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-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 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-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 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-sum 14417

This theorem is referenced by: dvnprodlem2 40162

W3C validator