dvnprodlem2 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem2 43014

Description: Induction step for dvnprodlem2 43014. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnprodlem2.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnprodlem2.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
dvnprodlem2.t	⊢ (𝜑 → 𝑇 ∈ Fin)
dvnprodlem2.h	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
dvnprodlem2.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
dvnprodlem2.dvnh	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
dvnprodlem2.c	⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
dvnprodlem2.r	⊢ (𝜑 → 𝑅 ⊆ 𝑇)
dvnprodlem2.z	⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅))
dvnprodlem2.ind	⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
dvnprodlem2.j	⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
dvnprodlem2.d	⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)

**Assertion**
Ref	Expression
dvnprodlem2	⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Distinct variable groups: 𝐶,𝑐,𝑘,𝑡 𝐷,𝑐,𝑡 𝐻,𝑐,𝑗,𝑘,𝑡 𝑥,𝐻,𝑐,𝑘,𝑡 𝐽,𝑐,𝑗,𝑘,𝑡 𝑛,𝐽,𝑠,𝑐,𝑘,𝑡 𝑥,𝐽 𝑗,𝑁,𝑡 𝑅,𝑐,𝑘,𝑛,𝑠,𝑡 𝑥,𝑅 𝑆,𝑐,𝑗,𝑘,𝑡 𝑥,𝑆 𝑇,𝑗,𝑡 𝑇,𝑠 𝑋,𝑐,𝑗,𝑘,𝑡 𝑥,𝑋 𝑍,𝑐,𝑗,𝑘,𝑡 𝑛,𝑍,𝑠 𝑥,𝑍 𝜑,𝑐,𝑗,𝑘,𝑡 𝜑,𝑛,𝑠 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥,𝑗,𝑛,𝑠) 𝐷(𝑥,𝑗,𝑘,𝑛,𝑠) 𝑅(𝑗) 𝑆(𝑛,𝑠) 𝑇(𝑥,𝑘,𝑛,𝑐) 𝐻(𝑛,𝑠) 𝑁(𝑥,𝑘,𝑛,𝑠,𝑐) 𝑋(𝑛,𝑠)

Proof of Theorem dvnprodlem2 Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1920	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ 𝑋)
2		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑡((𝐻‘𝑍)‘𝑥)
3		dvnprodlem2.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		dvnprodlem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ 𝑇)
5		ssfi 8765	. . . . . . . 8 ⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin)
6	3, 4, 5	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Fin)
7	6	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ Fin)
8		dvnprodlem2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅))
9	8	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ (𝑇 ∖ 𝑅))
10	8	eldifbd 3854	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
11	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑍 ∈ 𝑅)
12		simpl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝜑)
13	4	sselda 3875	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
14		dvnprodlem2.h	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
15	12, 13, 14	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
16	15	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
17		simplr 769	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
18	16, 17	ffvelrnd 6856	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
19		fveq2 6668	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐻‘𝑡) = (𝐻‘𝑍))
20	19	fveq1d 6670	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑍)‘𝑥))
21		id 22	. . . . . . . . 9 ⊢ (𝜑 → 𝜑)
22		eldifi 4015	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑇 ∖ 𝑅) → 𝑍 ∈ 𝑇)
23	8, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
24		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → 𝑍 ∈ 𝑇)
25		id 22	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝜑 ∧ 𝑍 ∈ 𝑇))
26		eleq1 2820	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑇 ↔ 𝑍 ∈ 𝑇))
27	26	anbi2d 632	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇)))
28	19	feq1d 6483	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡):𝑋⟶ℂ ↔ (𝐻‘𝑍):𝑋⟶ℂ))
29	27, 28	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)))
30	29, 14	vtoclg 3470	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ))
31	24, 25, 30	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)
32	21, 23, 31	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘𝑍):𝑋⟶ℂ)
33	32	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑍):𝑋⟶ℂ)
34		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
35	33, 34	ffvelrnd 6856	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐻‘𝑍)‘𝑥) ∈ ℂ)
36	1, 2, 7, 9, 11, 18, 20, 35	fprodsplitsn 15428	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥) = (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥)))
37	36	mpteq2dva 5122	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))
38	37	oveq2d 7180	. . 3 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥)))))
39	38	fveq1d 6670	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))‘𝐽))
40		dvnprodlem2.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
41		dvnprodlem2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
42	1, 7, 18	fprodclf 15431	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
43		dvnprodlem2.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
44		elfznn0 13084	. . . 4 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ∈ ℕ₀)
45	43, 44	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
46		eqid 2738	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
47		eqid 2738	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))
48		dvnprodlem2.c	. . . . . . . . . 10 ⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
49		oveq2 7172	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m 𝑅))
50		rabeq 3384	. . . . . . . . . . . . 13 ⊢ (((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})
51	49, 50	syl 17	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
52		sumeq1 15131	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
53	52	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛))
54	53	rabbidv 3380	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
55	51, 54	eqtrd 2773	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
56	55	mpteq2dv 5123	. . . . . . . . . 10 ⊢ (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}))
57		ssexg 5188	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → 𝑅 ∈ V)
58	4, 3, 57	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ V)
59		elpwg 4488	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
60	58, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
61	4, 60	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇)
62	61	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑅 ∈ 𝒫 𝑇)
63		nn0ex 11975	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
64	63	mptex 6990	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V)
66	48, 56, 62, 65	fvmptd3 6792	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
67		oveq2 7172	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
68	67	oveq1d 7179	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑_m 𝑅) = ((0...𝑘) ↑_m 𝑅))
69		rabeq 3384	. . . . . . . . . . . 12 ⊢ (((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})
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
71		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘))
72	71	rabbidv 3380	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
73	70, 72	eqtrd 2773	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
74	73	adantl 485	. . . . . . . . 9 ⊢ (((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})
75		elfznn0 13084	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ₀)
76	75	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ₀)
77		fzfid 13425	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑘) ∈ Fin)
78		mapfi 8886	. . . . . . . . . . . 12 ⊢ (((0...𝑘) ∈ Fin ∧ 𝑅 ∈ Fin) → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
79	77, 6, 78	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
80	79	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
81		ssrab2 3967	. . . . . . . . . . 11 ⊢ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑_m 𝑅)
82	81	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑_m 𝑅))
83	80, 82	ssexd 5189	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V)
84	66, 74, 76, 83	fvmptd 6776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
85		ssfi 8765	. . . . . . . . . 10 ⊢ ((((0...𝑘) ↑_m 𝑅) ∈ Fin ∧ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑_m 𝑅)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
86	79, 81, 85	sylancl 589	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
87	86	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
88	84, 87	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
89	88	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
90	75	faccld 13729	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℕ)
91	90	nncnd 11725	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℂ)
92	91	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (!‘𝑘) ∈ ℂ)
93	6	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
94	93	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
95		elfznn0 13084	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
96	95	ssriv 3879	. . . . . . . . . . . . . 14 ⊢ (0...𝑘) ⊆ ℕ₀
97	96	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℕ₀)
98		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))
99	84	eleq2d 2818	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}))
100	99	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}))
101	98, 100	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
102	81	sseli 3871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → 𝑐 ∈ ((0...𝑘) ↑_m 𝑅))
103	101, 102	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((0...𝑘) ↑_m 𝑅))
104		elmapi 8452	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ((0...𝑘) ↑_m 𝑅) → 𝑐:𝑅⟶(0...𝑘))
105	103, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐:𝑅⟶(0...𝑘))
106	105	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑐:𝑅⟶(0...𝑘))
107		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅)
108	106, 107	ffvelrnd 6856	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑘))
109	97, 108	sseldd 3876	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
110	109	faccld 13729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
111	110	nncnd 11725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
112	94, 111	fprodcl 15391	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
113	110	nnne0d 11759	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ≠ 0)
114	94, 111, 113	fprodn0 15418	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
115	92, 112, 114	divcld 11487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
116	115	adantlr 715	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
117	94	adantlr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
118	21	ad4antr 732	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
119	118, 13	sylancom 591	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
120		elfzuz3 12988	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ (ℤ_≥‘𝑘))
121		fzss2 13031	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (ℤ_≥‘𝑘) → (0...𝑘) ⊆ (0...𝐽))
122	120, 121	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝑘) ⊆ (0...𝐽))
123	122	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝐽))
124	45	nn0zd 12159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ ℤ)
125		dvnprodlem2.n	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
126	125	nn0zd 12159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
127		elfzle2 12995	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ≤ 𝑁)
128	43, 127	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
129	124, 126, 128	3jca 1129	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
130		eluz2 12323	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) ↔ (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
131	129, 130	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐽))
132		fzss2 13031	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) → (0...𝐽) ⊆ (0...𝑁))
133	131, 132	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝐽) ⊆ (0...𝑁))
134	133	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝐽) ⊆ (0...𝑁))
135	123, 134	sstrd 3885	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝑁))
136	135	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ (0...𝑁))
137	136, 108	sseldd 3876	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
138	137	adantllr 719	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
139		fvex 6681	. . . . . . . . . . 11 ⊢ (𝑐‘𝑡) ∈ V
140		eleq1 2820	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑡) ∈ (0...𝑁)))
141	140	3anbi3d 1443	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁))))
142		fveq2 6668	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
143	142	feq1d 6483	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑐‘𝑡) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ))
144	141, 143	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑗 = (𝑐‘𝑡) → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)))
145		dvnprodlem2.dvnh	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
146	139, 144, 145	vtocl 3463	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
147	118, 119, 138, 146	syl3anc 1372	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
148		simpllr 776	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
149	147, 148	ffvelrnd 6856	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
150	117, 149	fprodcl 15391	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
151	116, 150	mulcld 10732	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ)
152	89, 151	fsumcl 15176	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ)
153	152	fmpttd 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))):𝑋⟶ℂ)
154		dvnprodlem2.ind	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
155	154	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
156		0zd 12067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ)
157	126	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℤ)
158		elfzelz 12991	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
159	158	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
160		elfzle1 12994	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ 𝑘)
161	160	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
162	159	zred 12161	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
163	45	nn0red 12030	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℝ)
164	163	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
165	157	zred 12161	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℝ)
166		elfzle2 12995	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽)
167	166	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽)
168	128	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ≤ 𝑁)
169	162, 164, 165, 167, 168	letrd 10868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝑁)
170	156, 157, 159, 161, 169	elfzd 12982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ (0...𝑁))
171		rspa 3118	. . . . . 6 ⊢ ((∀𝑘 ∈ (0...𝑁)((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
172	155, 170, 171	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
173	172	feq1d 6483	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘):𝑋⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))):𝑋⟶ℂ))
174	153, 173	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘):𝑋⟶ℂ)
175	23	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑍 ∈ 𝑇)
176		simpl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝜑)
177	176, 175, 170	3jca 1129	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)))
178	26	3anbi2d 1442	. . . . . . 7 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
179	19	oveq2d 7180	. . . . . . . . 9 ⊢ (𝑡 = 𝑍 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑍)))
180	179	fveq1d 6670	. . . . . . . 8 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
181	180	feq1d 6483	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
182	178, 181	imbi12d 348	. . . . . 6 ⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)))
183		eleq1 2820	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...𝑁)))
184	183	3anbi3d 1443	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
185		fveq2 6668	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘))
186	185	feq1d 6483	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ))
187	184, 186	imbi12d 348	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)))
188	187, 145	chvarvv 2009	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)
189	182, 188	vtoclg 3470	. . . . 5 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
190	175, 177, 189	sylc 65	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)
191	32	feqmptd 6731	. . . . . . . . 9 ⊢ (𝜑 → (𝐻‘𝑍) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
192	191	eqcomd 2744	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝐻‘𝑍))
193	192	oveq2d 7180	. . . . . . 7 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
194	193	fveq1d 6670	. . . . . 6 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
195	194	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
196	195	feq1d 6483	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
197	190, 196	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ)
198		fveq2 6668	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑡)‘𝑥))
199	198	prodeq2ad 42659	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦) = ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
200	199	cbvmptv 5130	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
201	200	oveq2i 7175	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))
202	201	fveq1i 6669	. . . 4 ⊢ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘)
203	202	mpteq2i 5119	. . 3 ⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘))
204		fveq2 6668	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑍)‘𝑦) = ((𝐻‘𝑍)‘𝑥))
205	204	cbvmptv 5130	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))
206	205	oveq2i 7175	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
207	206	fveq1i 6669	. . . 4 ⊢ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘)
208	207	mpteq2i 5119	. . 3 ⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘))
209	40, 41, 42, 35, 45, 46, 47, 174, 197, 203, 208	dvnmul 43010	. 2 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥)))))
210	202	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘))
211	154	r19.21bi 3120	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
212	176, 170, 211	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
213	210, 212	eqtrd 2773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
214	213	mpteq2dva 5122	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘)) = (𝑘 ∈ (0...𝐽) ↦ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))))
215		mptexg 6988	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V)
216	41, 215	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V)
217	216	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) ∈ V)
218	214, 217	fvmpt2d 6782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
219	218	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
220	219	fveq1d 6670	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) = ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥))
221	34	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → 𝑥 ∈ 𝑋)
222	152	an32s 652	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ)
223		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
224	223	fvmpt2 6780	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋 ∧ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) → ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
225	221, 222, 224	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
226	220, 225	eqtrd 2773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
227		fveq2 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗))
228	227	cbvmptv 5130	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗))
229	228	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)))
230	206, 193	syl5eq 2785	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
231	230	fveq1d 6670	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
232	231	mpteq2dv 5123	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗)))
233	229, 232	eqtrd 2773	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗)))
234	233	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗)))
235		fveq2 6668	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
236	235	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑗 = (𝐽 − 𝑘)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
237		0zd 12067	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 0 ∈ ℤ)
238		elfzel2 12989	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
239	238, 158	zsubcld 12166	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ∈ ℤ)
240	237, 238, 239	3jca 1129	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ))
241	238	zred 12161	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
242	75	nn0red 12030	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
243	241, 242	subge0d 11301	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽))
244	166, 243	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘))
245	241, 242	subge02d 11303	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽))
246	160, 245	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ≤ 𝐽)
247	240, 244, 246	jca32 519	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝐽) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
248	247	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
249		elfz2 12981	. . . . . . . . . . . 12 ⊢ ((𝐽 − 𝑘) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
250	248, 249	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ∈ (0...𝐽))
251		fvex 6681	. . . . . . . . . . . 12 ⊢ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V
252	251	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V)
253	234, 236, 250, 252	fvmptd 6776	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘)) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
254	253	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘)) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
255	254	fveq1d 6670	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))
256	226, 255	oveq12d 7182	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥)) = (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))
257	256	oveq2d 7180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = ((𝐽C𝑘) · (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
258	88	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
259		ovex 7197	. . . . . . . . . . . 12 ⊢ (𝐽 − 𝑘) ∈ V
260		eleq1 2820	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − 𝑘) → (𝑗 ∈ (0...𝐽) ↔ (𝐽 − 𝑘) ∈ (0...𝐽)))
261	260	anbi2d 632	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽))))
262	235	feq1d 6483	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − 𝑘) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ))
263	261, 262	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐽 − 𝑘) → (((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)))
264		eleq1 2820	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝐽) ↔ 𝑗 ∈ (0...𝐽)))
265	264	anbi2d 632	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝐽))))
266		fveq2 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
267	266	feq1d 6483	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
268	265, 267	imbi12d 348	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)))
269	268, 190	chvarvv 2009	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
270	259, 263, 269	vtocl 3463	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
271	176, 250, 270	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
272	271	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
273	272, 221	ffvelrnd 6856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
274		anass 472	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)))
275		ancom 464	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽)))
276	275	anbi2i 626	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))))
277		anass 472	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))))
278	277	bicomi 227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
279	276, 278	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
280	274, 279	bitri 278	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
281	280	anbi1i 627	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)))
282	281	imbi1i 353	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ) ↔ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ))
283	151, 282	mpbi 233	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) ∈ ℂ)
284	258, 273, 283	fsummulc1 15226	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))
285	284	oveq2d 7180	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · (Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)(((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = ((𝐽C𝑘) · Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
286	176, 45	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℕ₀)
287	286, 159	bccld 42376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℕ₀)
288	287	nn0cnd 12031	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
289	288	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
290	273	adantr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
291	283, 290	mulcld 10732	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) ∈ ℂ)
292	258, 289, 291	fsummulc2 15225	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
293	257, 285, 292	3eqtrd 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
294	293	sumeq2dv 15146	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑘 ∈ (0...𝐽)Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
295		vex 3401	. . . . . . . 8 ⊢ 𝑘 ∈ V
296		vex 3401	. . . . . . . 8 ⊢ 𝑐 ∈ V
297	295, 296	op1std 7717	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (1^st ‘𝑝) = 𝑘)
298	297	oveq2d 7180	. . . . . 6 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C𝑘))
299	297	fveq2d 6672	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘(1^st ‘𝑝)) = (!‘𝑘))
300	295, 296	op2ndd 7718	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (2^nd ‘𝑝) = 𝑐)
301	300	fveq1d 6670	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((2^nd ‘𝑝)‘𝑡) = (𝑐‘𝑡))
302	301	fveq2d 6672	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘(𝑐‘𝑡)))
303	302	prodeq2ad 42659	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
304	299, 303	oveq12d 7182	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
305	301	fveq2d 6672	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
306	305	fveq1d 6670	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
307	306	prodeq2ad 42659	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
308	304, 307	oveq12d 7182	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) = (((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
309	297	oveq2d 7180	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
310	309	fveq2d 6672	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
311	310	fveq1d 6670	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))
312	308, 311	oveq12d 7182	. . . . . 6 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥)) = ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)))
313	298, 312	oveq12d 7182	. . . . 5 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))) = ((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))))
314		fzfid 13425	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0...𝐽) ∈ Fin)
315	289	adantrr 717	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → (𝐽C𝑘) ∈ ℂ)
316	291	anasss 470	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥)) ∈ ℂ)
317	315, 316	mulcld 10732	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → ((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) ∈ ℂ)
318	313, 314, 258, 317	fsum2d 15212	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)Σ𝑐 ∈ ((𝐶‘𝑅)‘𝑘)((𝐽C𝑘) · ((((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))))
319		ovex 7197	. . . . . . . . 9 ⊢ (𝐽 − (𝑐‘𝑍)) ∈ V
320	296	resex 5867	. . . . . . . . 9 ⊢ (𝑐 ↾ 𝑅) ∈ V
321	319, 320	op1std 7717	. . . . . . . 8 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (1^st ‘𝑝) = (𝐽 − (𝑐‘𝑍)))
322	321	oveq2d 7180	. . . . . . 7 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C(𝐽 − (𝑐‘𝑍))))
323	321	fveq2d 6672	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘(1^st ‘𝑝)) = (!‘(𝐽 − (𝑐‘𝑍))))
324	319, 320	op2ndd 7718	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (2^nd ‘𝑝) = (𝑐 ↾ 𝑅))
325	324	fveq1d 6670	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((2^nd ‘𝑝)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
326	325	fveq2d 6672	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘((𝑐 ↾ 𝑅)‘𝑡)))
327	326	prodeq2ad 42659	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)))
328	323, 327	oveq12d 7182	. . . . . . . . 9 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))))
329	325	fveq2d 6672	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)))
330	329	fveq1d 6670	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥))
331	330	prodeq2ad 42659	. . . . . . . . 9 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥))
332	328, 331	oveq12d 7182	. . . . . . . 8 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)))
333	321	oveq2d 7180	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
334	333	fveq2d 6672	. . . . . . . . 9 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍)))))
335	334	fveq1d 6670	. . . . . . . 8 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))
336	332, 335	oveq12d 7182	. . . . . . 7 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥)) = ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)))
337	322, 336	oveq12d 7182	. . . . . 6 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))) = ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))))
338		oveq2 7172	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})))
339		rabeq 3384	. . . . . . . . . . . . 13 ⊢ (((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})
340	338, 339	syl 17	. . . . . . . . . . . 12 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
341		sumeq1 15131	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
342	341	eqeq1d 2740	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛))
343	342	rabbidv 3380	. . . . . . . . . . . 12 ⊢ (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})
344	340, 343	eqtrd 2773	. . . . . . . . . . 11 ⊢ (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})
345	344	mpteq2dv 5123	. . . . . . . . . 10 ⊢ (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}))
346	23	snssd 4694	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ⊆ 𝑇)
347	4, 346	unssd 4074	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
348	3, 347	ssexd 5189	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
349		elpwg 4488	. . . . . . . . . . . 12 ⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
350	348, 349	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
351	347, 350	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
352	63	mptex 6990	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V
353	352	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V)
354	48, 345, 351, 353	fvmptd3 6792	. . . . . . . . 9 ⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
355		oveq2 7172	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
356	355	oveq1d 7179	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
357		rabeq 3384	. . . . . . . . . . . 12 ⊢ (((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})
358	356, 357	syl 17	. . . . . . . . . . 11 ⊢ (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})
359		eqeq2 2750	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
360	359	rabbidv 3380	. . . . . . . . . . 11 ⊢ (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})
361	358, 360	eqtrd 2773	. . . . . . . . . 10 ⊢ (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})
362	361	adantl 485	. . . . . . . . 9 ⊢ ((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})
363		ovex 7197	. . . . . . . . . . 11 ⊢ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ V
364	363	rabex 5197	. . . . . . . . . 10 ⊢ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V
365	364	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V)
366	354, 362, 45, 365	fvmptd 6776	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
367		fzfid 13425	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝐽) ∈ Fin)
368		snfi 8635	. . . . . . . . . . . 12 ⊢ {𝑍} ∈ Fin
369	368	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑍} ∈ Fin)
370		unfi 8764	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ Fin)
371	6, 369, 370	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ Fin)
372		mapfi 8886	. . . . . . . . . 10 ⊢ (((0...𝐽) ∈ Fin ∧ (𝑅 ∪ {𝑍}) ∈ Fin) → ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ Fin)
373	367, 371, 372	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ Fin)
374		ssrab2 3967	. . . . . . . . . 10 ⊢ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍}))
375	374	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
376		ssfi 8765	. . . . . . . . 9 ⊢ ((((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ Fin ∧ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍}))) → {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin)
377	373, 375, 376	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin)
378	366, 377	eqeltrd 2833	. . . . . . 7 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin)
379	378	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin)
380		dvnprodlem2.d	. . . . . . . 8 ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
381	48, 45, 380, 3, 23, 10, 347	dvnprodlem1 43013	. . . . . . 7 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
382	381	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
383		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
384		opex 5319	. . . . . . . . 9 ⊢ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V
385	384	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V)
386	380	fvmpt2 6780	. . . . . . . 8 ⊢ ((𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
387	383, 385, 386	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
388	387	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
389	45	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℕ₀)
390		eliun 4882	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
391	390	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
392	391	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
393		nfv 1920	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝜑
394		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑝
395		nfiu1 4912	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
396	394, 395	nfel 2913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
397	393, 396	nfan 1905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
398		nfv 1920	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(1^st ‘𝑝) ∈ (0...𝐽)
399		xp1st 7739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ {𝑘})
400		elsni 4530	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑝) ∈ {𝑘} → (1^st ‘𝑝) = 𝑘)
401	399, 400	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) = 𝑘)
402	401	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
403		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
404	402, 403	eqeltrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
405	404	ex 416	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
406	405	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽))))
407	397, 398, 406	rexlimd 3226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
408	392, 407	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
409		elfzelz 12991	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℤ)
410	408, 409	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
411	389, 410	bccld 42376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℕ₀)
412	411	nn0cnd 12031	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
413	412	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
414		elfznn0 13084	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℕ₀)
415	408, 414	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℕ₀)
416	415	faccld 13729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℕ)
417	416	nncnd 11725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
418	417	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
419	6	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
420		nfv 1920	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(2^nd ‘𝑝):𝑅⟶(0...𝐽)
421	84, 82	eqsstrd 3913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝑘) ↑_m 𝑅))
422		ovex 7197	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐽) ∈ V
423	422	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝐽) ∈ V)
424		mapss 8492	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝐽) ∈ V ∧ (0...𝑘) ⊆ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
425	423, 122, 424	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
426	425	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
427	421, 426	sstrd 3885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_m 𝑅))
428	427	3adant3 1133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_m 𝑅))
429		xp2nd 7740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
430	429	3ad2ant3 1136	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
431	428, 430	sseldd 3876	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((0...𝐽) ↑_m 𝑅))
432		elmapi 8452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑝) ∈ ((0...𝐽) ↑_m 𝑅) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
433	431, 432	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
434	433	3exp 1120	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
435	434	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
436	397, 420, 435	rexlimd 3226	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽)))
437	392, 436	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
438	437	ffvelrnda 6855	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽))
439		elfznn0 13084	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((2^nd ‘𝑝)‘𝑡) ∈ ℕ₀)
440	439	faccld 13729	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
441	440	nncnd 11725	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
442	438, 441	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
443	419, 442	fprodcl 15391	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
444	443	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
445	438, 440	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
446		nnne0 11743	. . . . . . . . . . . . 13 ⊢ ((!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
447	445, 446	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
448	419, 442, 447	fprodn0 15418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
449	448	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
450	418, 444, 449	divcld 11487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) ∈ ℂ)
451	7	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
452		simpll 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝜑)
453	452, 13	sylancom 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
454	452, 133	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
455	454, 438	sseldd 3876	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))
456	452, 453, 455	3jca 1129	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
457		eleq1 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
458	457	3anbi3d 1443	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))))
459		fveq2 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)))
460	459	feq1d 6483	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ))
461	458, 460	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)))
462	461, 145	vtoclg 3470	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ))
463	438, 456, 462	sylc 65	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)
464	463	adantllr 719	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)
465	17	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
466	464, 465	ffvelrnd 6856	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) ∈ ℂ)
467	451, 466	fprodcl 15391	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) ∈ ℂ)
468	450, 467	mulcld 10732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) ∈ ℂ)
469		nfv 1920	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)
470		simp1 1137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
471	404	3adant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
472		fznn0sub2 13098	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
473	472	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ (0...𝐽)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
474	470, 471, 473	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
475	474	3exp 1120	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
476	475	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
477	397, 469, 476	rexlimd 3226	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)))
478	392, 477	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
479		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
480	479, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇)
481	479, 133	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ⊆ (0...𝑁))
482	481, 478	sseldd 3876	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))
483	479, 480, 482	3jca 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
484		eleq1 2820	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → (𝑗 ∈ (0...𝑁) ↔ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
485	484	3anbi3d 1443	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))))
486		fveq2 6668	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))))
487	486	feq1d 6483	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ))
488	485, 487	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → (((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)))
489		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → 𝑍 ∈ 𝑇)
490		id 22	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)))
491	26	3anbi2d 1442	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁))))
492	179	fveq1d 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
493	492	feq1d 6483	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
494	491, 493	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)))
495	494, 145	vtoclg 3470	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
496	489, 490, 495	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
497	488, 496	vtoclg 3470	. . . . . . . . . . 11 ⊢ ((𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ))
498	478, 483, 497	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
499	498	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
500	34	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑥 ∈ 𝑋)
501	499, 500	ffvelrnd 6856	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) ∈ ℂ)
502	468, 501	mulcld 10732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥)) ∈ ℂ)
503	413, 502	mulcld 10732	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))) ∈ ℂ)
504	337, 379, 382, 388, 503	fsumf1o 15166	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))))
505		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝜑)
506	366	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
507	383, 506	eleqtrd 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
508	374	sseli 3871	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} → 𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
509	507, 508	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
510		elmapi 8452	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
511	509, 510	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
512		snidg 4547	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍})
513	23, 512	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
514		elun2 4065	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
515	513, 514	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍}))
516	515	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
517	511, 516	ffvelrnd 6856	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽))
518		0zd 12067	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ∈ ℤ)
519	124	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℤ)
520		fzssz 12993	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℤ
521	520	sseli 3871	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℤ)
522	521	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℤ)
523	519, 522	zsubcld 12166	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
524		elfzle2 12995	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽)
525	524	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ≤ 𝐽)
526	163	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
527	522	zred 12161	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℝ)
528	526, 527	subge0d 11301	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽))
529	525, 528	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍)))
530		elfzle1 12994	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍))
531	530	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝑐‘𝑍))
532	526, 527	subge02d 11303	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))
533	531, 532	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)
534	518, 519, 523, 529, 533	elfzd 12982	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
535	505, 517, 534	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
536		bcval2 13750	. . . . . . . . . . 11 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
537	535, 536	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
538	163	recnd 10740	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℂ)
539	538	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
540		zsscn 12063	. . . . . . . . . . . . . . . . 17 ⊢ ℤ ⊆ ℂ
541	520, 540	sstri 3884	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℂ
542	541	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
543	542, 517	sseldd 3876	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ)
544	539, 543	nncand 11073	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍))
545	544	fveq2d 6672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = (!‘(𝑐‘𝑍)))
546	545	oveq1d 7179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍)))))
547	546	oveq2d 7180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
548	45	faccld 13729	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘𝐽) ∈ ℕ)
549	548	nncnd 11725	. . . . . . . . . . . . 13 ⊢ (𝜑 → (!‘𝐽) ∈ ℂ)
550	549	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
551		elfznn0 13084	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℕ₀)
552	517, 551	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℕ₀)
553	552	faccld 13729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℕ)
554	553	nncnd 11725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
555		elfznn0 13084	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
556	535, 555	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
557	556	faccld 13729	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℕ)
558	557	nncnd 11725	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
559	553	nnne0d 11759	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
560	557	nnne0d 11759	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
561	550, 554, 558, 559, 560	divdiv1d 11518	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
562	561	eqcomd 2744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
563	537, 547, 562	3eqtrd 2777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
564	563	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
565		fvres 6687	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
566	565	fveq2d 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → (!‘((𝑐 ↾ 𝑅)‘𝑡)) = (!‘(𝑐‘𝑡)))
567	566	prodeq2i 15358	. . . . . . . . . . . . . . 15 ⊢ ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))
568	567	oveq2i 7175	. . . . . . . . . . . . . 14 ⊢ ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
569	565	fveq2d 6672	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
570	569	fveq1d 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑅 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
571	570	prodeq2i 15358	. . . . . . . . . . . . . 14 ⊢ ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)
572	568, 571	oveq12i 7176	. . . . . . . . . . . . 13 ⊢ (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
573	572	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
574	573	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
575	558	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
576	505, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
577	75	ssriv 3879	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐽) ⊆ ℕ₀
578	577	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℕ₀)
579	511	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
580		elun1 4064	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝑅 → 𝑡 ∈ (𝑅 ∪ {𝑍}))
581	580	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
582	579, 581	ffvelrnd 6856	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽))
583	578, 582	sseldd 3876	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
584	583	faccld 13729	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
585	584	nncnd 11725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
586	576, 585	fprodcl 15391	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
587	586	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
588	7	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
589	505	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
590	505, 13	sylan 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
591	589, 133	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
592	591, 582	sseldd 3876	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
593	589, 590, 592, 146	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
594	593	adantllr 719	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
595	17	adantlr 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
596	594, 595	ffvelrnd 6856	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
597	588, 596	fprodcl 15391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
598	576, 584	fprodnncl 15394	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ)
599		nnne0 11743	. . . . . . . . . . . . . 14 ⊢ (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
600	598, 599	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
601	600	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
602	575, 587, 597, 601	div32d 11510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))))
603	574, 602	eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))))
604	544	fveq2d 6672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
605	604	fveq1d 6670	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))
606	605	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))
607	603, 606	oveq12d 7182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)) = (((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))
608	597, 587, 601	divcld 11487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
609	505, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
610	505, 133	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ (0...𝑁))
611	610, 517	sseldd 3876	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝑁))
612	505, 609, 611	3jca 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)))
613		eleq1 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑍) ∈ (0...𝑁)))
614	613	3anbi3d 1443	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁))))
615		fveq2 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
616	615	feq1d 6483	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
617	614, 616	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑐‘𝑍) → (((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)))
618	617, 496	vtoclg 3470	. . . . . . . . . . . . 13 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
619	517, 612, 618	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
620	619	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
621	34	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑥 ∈ 𝑋)
622	620, 621	ffvelrnd 6856	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥) ∈ ℂ)
623	575, 608, 622	mulassd 10735	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘(𝐽 − (𝑐‘𝑍))) · (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))))
624	607, 623	eqtrd 2773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥)) = ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))))
625	564, 624	oveq12d 7182	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = ((((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) · ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))))
626	549	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
627	554	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
628	559	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
629	626, 627, 628	divcld 11487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / (!‘(𝑐‘𝑍))) ∈ ℂ)
630	608, 622	mulcld 10732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) ∈ ℂ)
631	560	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
632	629, 575, 630, 631	dmmcand 42374	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) · ((!‘(𝐽 − (𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))))
633	597, 622, 587, 601	div23d 11524	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))
634	633	eqcomd 2744	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
635		nfv 1920	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
636		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)
637	609	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
638	11	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
639		fveq2 6668	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (𝑐‘𝑡) = (𝑐‘𝑍))
640	179, 639	fveq12d 6675	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
641	640	fveq1d 6670	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))
642	635, 636, 588, 637, 638, 596, 641, 622	fprodsplitsn 15428	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)))
643	642	eqcomd 2744	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
644	643	oveq1d 7179	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
645	634, 644	eqtrd 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)) = (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
646	645	oveq2d 7180	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) · (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))))
647	588, 368, 370	sylancl 589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
648	505	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝜑)
649	347	sselda 3875	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
650	649	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
651	511, 610	fssd 6516	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝑁))
652	651	ffvelrnda 6855	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝑁))
653	648, 650, 652, 146	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
654	653	adantllr 719	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
655	621	adantr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑥 ∈ 𝑋)
656	654, 655	ffvelrnd 6856	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
657	647, 656	fprodcl 15391	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
658	626, 627, 657, 587, 628, 601	divmuldivd 11528	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · (∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))))
659	554, 586	mulcomd 10733	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
660		nfv 1920	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
661		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(!‘(𝑐‘𝑍))
662	505, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
663	639	fveq2d 6672	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑍)))
664	660, 661, 576, 609, 662, 585, 663, 554	fprodsplitsn 15428	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
665	664	eqcomd 2744	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
666	659, 665	eqtrd 2773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
667	666	oveq2d 7180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))))
668	667	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))))
669	505, 371	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
670	577	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0...𝐽) ⊆ ℕ₀)
671	511	ffvelrnda 6855	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝐽))
672	670, 671	sseldd 3876	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ ℕ₀)
673	672	faccld 13729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℕ)
674	673	nncnd 11725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℂ)
675	669, 674	fprodcl 15391	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
676	675	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
677	673	nnne0d 11759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ≠ 0)
678	669, 674, 677	fprodn0 15418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
679	678	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
680	626, 657, 676, 679	div23d 11524	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
681		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
682	668, 680, 681	3eqtrd 2777	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)) / ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
683	646, 658, 682	3eqtrd 2777	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) · ((∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
684	625, 632, 683	3eqtrd 2777	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = (((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
685	684	sumeq2dv 15146	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐽C(𝐽 − (𝑐‘𝑍))) · ((((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍))))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
686	504, 685	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))((𝐽C(1^st ‘𝑝)) · ((((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) · ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥)) · (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
687	294, 318, 686	3eqtrd 2777	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥))) = Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)))
688	687	mpteq2dva 5122	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ (0...𝐽)((𝐽C𝑘) · ((((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘))‘𝑘)‘𝑥) · (((𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘))‘(𝐽 − 𝑘))‘𝑥)))) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))
689	39, 209, 688	3eqtrd 2777	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)))‘𝐽) = (𝑥 ∈ 𝑋 ↦ Σ𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)(((!‘𝐽) / ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡))) · ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 ∀wral 3053 ∃wrex 3054 {crab 3057 Vcvv 3397 ∖ cdif 3838 ∪ cun 3839 ⊆ wss 3841 𝒫 cpw 4485 {csn 4513 {cpr 4515 ⟨cop 4519 ∪ ciun 4878 class class class wbr 5027 ↦ cmpt 5107 × cxp 5517 ↾ cres 5521 ⟶wf 6329 –1-1-onto→wf1o 6332 ‘cfv 6333 (class class class)co 7164 1^st c1st 7705 2^nd c2nd 7706 ↑_m cmap 8430 Fincfn 8548 ℂcc 10606 ℝcr 10607 0cc0 10608 · cmul 10613 ≤ cle 10747 − cmin 10941 / cdiv 11368 ℕcn 11709 ℕ₀cn0 11969 ℤcz 12055 ℤ_≥cuz 12317 ...cfz 12974 !cfa 13718 Ccbc 13747 Σcsu 15128 ∏cprod 15344 ↾_t crest 16790 TopOpenctopn 16791 ℂ_fldccnfld 20210 D^𝑛 cdvn 24608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-fac 13719 df-bc 13748 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-sum 15129 df-prod 15345 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-mulg 18336 df-cntz 18558 df-cmn 19019 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-fbas 20207 df-fg 20208 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-lp 21880 df-perf 21881 df-cn 21971 df-cnp 21972 df-haus 22059 df-tx 22306 df-hmeo 22499 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-xms 23066 df-ms 23067 df-tms 23068 df-cncf 23623 df-limc 24610 df-dv 24611 df-dvn 24612

This theorem is referenced by: dvnprodlem3 43015

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