dvnprodlem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvnprodlem2		Structured version Visualization version GIF version

Theorem dvnprodlem2 46588

Description: Induction step for dvnprodlem2 46588. (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 1941	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ 𝑋)
2		nfcv 2931	. . . . . 6 ⊢ Ⅎ𝑡((𝐻‘𝑍)‘𝑥)
3		dvnprodlem2.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		dvnprodlem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ 𝑇)
5		ssfi 9157	. . . . . . . 8 ⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin)
6	3, 4, 5	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Fin)
7	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ Fin)
8		dvnprodlem2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅))
9	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ (𝑇 ∖ 𝑅))
10	8	eldifbd 3924	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
11	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑍 ∈ 𝑅)
12		simpl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝜑)
13	4	sselda 3943	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
14		dvnprodlem2.h	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
15	12, 13, 14	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
16	15	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
17		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
18	16, 17	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
19		fveq2 6882	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐻‘𝑡) = (𝐻‘𝑍))
20	19	fveq1d 6884	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑍)‘𝑥))
21		id 23	. . . . . . . . 9 ⊢ (𝜑 → 𝜑)
22		eldifi 4091	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑇 ∖ 𝑅) → 𝑍 ∈ 𝑇)
23	8, 22	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
24		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → 𝑍 ∈ 𝑇)
25		id 23	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝜑 ∧ 𝑍 ∈ 𝑇))
26		eleq1 2857	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑇 ↔ 𝑍 ∈ 𝑇))
27	26	anbi2d 641	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇)))
28	19	feq1d 6688	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡):𝑋⟶ℂ ↔ (𝐻‘𝑍):𝑋⟶ℂ))
29	27, 28	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)))
30	29, 14	vtoclg 3529	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ))
31	24, 25, 30	sylc 66	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)
32	21, 23, 31	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘𝑍):𝑋⟶ℂ)
33	32	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑍):𝑋⟶ℂ)
34		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
35	33, 34	ffvelcdmd 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐻‘𝑍)‘𝑥) ∈ ℂ)
36	1, 2, 7, 9, 11, 18, 20, 35	fprodsplitsn 16043	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥) = (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥)))
37	36	mpteq2dva 5206	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))
38	37	oveq2d 7427	. . 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 6884	. 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 16046	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
43		dvnprodlem2.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
44		elfznn0 13648	. . . 4 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ∈ ℕ₀)
45	43, 44	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
46		eqid 2769	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
47		eqid 2769	. . 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 7419	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m 𝑅))
50		rabeq 3436	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 15740	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
53	52	eqeq1d 2771	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛))
54	53	rabbidv 3429	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 5207	. . . . . . . . . 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 5294	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → 𝑅 ∈ V)
58	4, 3, 57	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ V)
59		elpwg 4568	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
60	58, 59	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
61	4, 60	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇)
62	61	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑅 ∈ 𝒫 𝑇)
63		nn0ex 12510	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
64	63	mptex 7222	. . . . . . . . . . 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 7014	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
67		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
68	67	oveq1d 7426	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑_m 𝑅) = ((0...𝑘) ↑_m 𝑅))
69		rabeq 3436	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 2781	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘))
72	71	rabbidv 3429	. . . . . . . . . . 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 2804	. . . . . . . . . 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 486	. . . . . . . . 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 13648	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ₀)
76	75	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ₀)
77		fzfid 14009	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑘) ∈ Fin)
78		mapfi 9305	. . . . . . . . . . . 12 ⊢ (((0...𝑘) ∈ Fin ∧ 𝑅 ∈ Fin) → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
79	77, 6, 78	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
80	79	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ∈ Fin)
81		ssrab2 4040	. . . . . . . . . . 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 5295	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V)
84	66, 74, 76, 83	fvmptd 6998	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
85		ssfi 9157	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
87	86	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
88	84, 87	eqeltrd 2869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
89	88	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
90	75	faccld 14320	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℕ)
91	90	nncnd 12249	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℂ)
92	91	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (!‘𝑘) ∈ ℂ)
93	6	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
94	93	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
95		elfznn0 13648	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
96	95	ssriv 3947	. . . . . . . . . . . . . 14 ⊢ (0...𝑘) ⊆ ℕ₀
97	96	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℕ₀)
98		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))
99	84	eleq2d 2855	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}))
100	99	adantr 485	. . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → 𝑐 ∈ ((0...𝑘) ↑_m 𝑅))
103	101, 102	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((0...𝑘) ↑_m 𝑅))
104		elmapi 8846	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ((0...𝑘) ↑_m 𝑅) → 𝑐:𝑅⟶(0...𝑘))
105	103, 104	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐:𝑅⟶(0...𝑘))
106	105	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑐:𝑅⟶(0...𝑘))
107		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅)
108	106, 107	ffvelcdmd 7081	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑘))
109	97, 108	sseldd 3944	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
110	109	faccld 14320	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
111	110	nncnd 12249	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
112	94, 111	fprodcl 16006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
113	110	nnne0d 12286	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ≠ 0)
114	94, 111, 113	fprodn0 16033	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
115	92, 112, 114	divcld 11991	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
116	115	adantlr 727	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
117	94	adantlr 727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
118	21	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
119	118, 13	sylancom 599	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
120		elfzuz3 13549	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ (ℤ_≥‘𝑘))
121		fzss2 13592	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (ℤ_≥‘𝑘) → (0...𝑘) ⊆ (0...𝐽))
122	120, 121	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝑘) ⊆ (0...𝐽))
123	122	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝐽))
124	45	nn0zd 12616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ ℤ)
125		dvnprodlem2.n	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
126	125	nn0zd 12616	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
127		elfzle2 13556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ≤ 𝑁)
128	43, 127	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
129	124, 126, 128	3jca 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
130		eluz2 12868	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) ↔ (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
131	129, 130	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐽))
132		fzss2 13592	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) → (0...𝐽) ⊆ (0...𝑁))
133	131, 132	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝐽) ⊆ (0...𝑁))
134	133	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝐽) ⊆ (0...𝑁))
135	123, 134	sstrd 3953	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝑁))
136	135	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ (0...𝑁))
137	136, 108	sseldd 3944	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
138	137	adantllr 731	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
139		fvex 6895	. . . . . . . . . . 11 ⊢ (𝑐‘𝑡) ∈ V
140		eleq1 2857	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑡) ∈ (0...𝑁)))
141	140	3anbi3d 1468	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁))))
142		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
143	142	feq1d 6688	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3532	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
147	118, 119, 138, 146	syl3anc 1396	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
148		simpllr 787	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
149	147, 148	ffvelcdmd 7081	. . . . . . . 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 16006	. . . . . . 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 11229	. . . . . 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 15784	. . . . 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 7111	. . . 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 485	. . . . . 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 12603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ)
157	126	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℤ)
158		elfzelz 13552	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
159	158	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
160		elfzle1 13555	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ 𝑘)
161	160	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
162	159	zred 12700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
163	45	nn0red 12566	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ ℝ)
164	163	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
165	157	zred 12700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℝ)
166		elfzle2 13556	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽)
167	166	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽)
168	128	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ≤ 𝑁)
169	162, 164, 165, 167, 168	letrd 11367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝑁)
170	156, 157, 159, 161, 169	elfzd 13543	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ (0...𝑁))
171		rspa 3260	. . . . . 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 595	. . . . 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 6688	. . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑍 ∈ 𝑇)
176		simpl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝜑)
177	176, 175, 170	3jca 1144	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)))
178	26	3anbi2d 1467	. . . . . . 7 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
179	19	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑡 = 𝑍 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑍)))
180	179	fveq1d 6884	. . . . . . . 8 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
181	180	feq1d 6688	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
182	178, 181	imbi12d 347	. . . . . 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 2857	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...𝑁)))
184	183	3anbi3d 1468	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
185		fveq2 6882	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘))
186	185	feq1d 6688	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ))
187	184, 186	imbi12d 347	. . . . . . 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 2016	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)
189	182, 188	vtoclg 3529	. . . . 5 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
190	175, 177, 189	sylc 66	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)
191	32	feqmptd 6950	. . . . . . . . 9 ⊢ (𝜑 → (𝐻‘𝑍) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
192	191	eqcomd 2775	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝐻‘𝑍))
193	192	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
194	193	fveq1d 6884	. . . . . 6 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
195	194	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
196	195	feq1d 6688	. . . 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 6882	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑡)‘𝑥))
199	198	prodeq2ad 46235	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦) = ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
200	199	cbvmptv 5217	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
201	200	oveq2i 7422	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))
202	201	fveq1i 6883	. . . 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 5209	. . 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 6882	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑍)‘𝑦) = ((𝐻‘𝑍)‘𝑥))
205	204	cbvmptv 5217	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))
206	205	oveq2i 7422	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
207	206	fveq1i 6883	. . . 4 ⊢ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘)
208	207	mpteq2i 5209	. . 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 46584	. 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 3263	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . 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 5206	. . . . . . . . . . . 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 7220	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 7004	. . . . . . . . . . 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 727	. . . . . . . . . 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 6884	. . . . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → 𝑥 ∈ 𝑋)
222	152	an32s 664	. . . . . . . . . 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 2769	. . . . . . . . . . 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 7002	. . . . . . . . . 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 595	. . . . . . . . 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 2804	. . . . . . . 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 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗))
228	227	cbvmptv 5217	. . . . . . . . . . . . . 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	eqtrid 2816	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
231	230	fveq1d 6884	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
232	231	mpteq2dv 5207	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
236	235	adantl 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑗 = (𝐽 − 𝑘)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
237		0zd 12603	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 0 ∈ ℤ)
238		elfzel2 13550	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
239	238, 158	zsubcld 12705	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ∈ ℤ)
240	237, 238, 239	3jca 1144	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ))
241	238	zred 12700	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
242	75	nn0red 12566	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
243	241, 242	subge0d 11804	. . . . . . . . . . . . . . 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 11806	. . . . . . . . . . . . . . 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 524	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝐽) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
248	247	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
249		elfz2 13542	. . . . . . . . . . . 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 6895	. . . . . . . . . . . 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 6998	. . . . . . . . . 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 727	. . . . . . . . 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 6884	. . . . . . . 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 7429	. . . . . . 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 7427	. . . . . 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 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
259		ovex 7444	. . . . . . . . . . . 12 ⊢ (𝐽 − 𝑘) ∈ V
260		eleq1 2857	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − 𝑘) → (𝑗 ∈ (0...𝐽) ↔ (𝐽 − 𝑘) ∈ (0...𝐽)))
261	260	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽))))
262	235	feq1d 6688	. . . . . . . . . . . . 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 347	. . . . . . . . . . . 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 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝐽) ↔ 𝑗 ∈ (0...𝐽)))
265	264	anbi2d 641	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝐽))))
266		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
267	266	feq1d 6688	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
268	265, 267	imbi12d 347	. . . . . . . . . . . . 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 2016	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
270	259, 263, 269	vtocl 3532	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
271	176, 250, 270	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
272	271	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
273	272, 221	ffvelcdmd 7081	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
274		anass 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)))
275		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽)))
276	275	anbi2i 634	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))))
277		anass 473	. . . . . . . . . . . . . 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 635	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)))
282	281	imbi1i 352	. . . . . . . . 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 15836	. . . . . . 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 7427	. . . . . 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 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℕ₀)
287	286, 159	bccld 45961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℕ₀)
288	287	nn0cnd 12567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
289	288	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
290	273	adantr 485	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
291	283, 290	mulcld 11229	. . . . . . 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 15835	. . . . . 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 2808	. . . . 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 15753	. . . 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 3465	. . . . . . . 8 ⊢ 𝑘 ∈ V
296		vex 3465	. . . . . . . 8 ⊢ 𝑐 ∈ V
297	295, 296	op1std 7996	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (1^st ‘𝑝) = 𝑘)
298	297	oveq2d 7427	. . . . . 6 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C𝑘))
299	297	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘(1^st ‘𝑝)) = (!‘𝑘))
300	295, 296	op2ndd 7997	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (2^nd ‘𝑝) = 𝑐)
301	300	fveq1d 6884	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((2^nd ‘𝑝)‘𝑡) = (𝑐‘𝑡))
302	301	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘(𝑐‘𝑡)))
303	302	prodeq2ad 46235	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
304	299, 303	oveq12d 7429	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
305	301	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
306	305	fveq1d 6884	. . . . . . . . 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 46235	. . . . . . . 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 7429	. . . . . . 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 7427	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
310	309	fveq2d 6886	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
311	310	fveq1d 6884	. . . . . . 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 7429	. . . . . 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 7429	. . . . 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 14009	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0...𝐽) ∈ Fin)
315	289	adantrr 729	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → (𝐽C𝑘) ∈ ℂ)
316	291	anasss 471	. . . . . 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 11229	. . . . 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 15822	. . . 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 7444	. . . . . . . . 9 ⊢ (𝐽 − (𝑐‘𝑍)) ∈ V
320	296	resex 6029	. . . . . . . . 9 ⊢ (𝑐 ↾ 𝑅) ∈ V
321	319, 320	op1std 7996	. . . . . . . 8 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (1^st ‘𝑝) = (𝐽 − (𝑐‘𝑍)))
322	321	oveq2d 7427	. . . . . . 7 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C(𝐽 − (𝑐‘𝑍))))
323	321	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘(1^st ‘𝑝)) = (!‘(𝐽 − (𝑐‘𝑍))))
324	319, 320	op2ndd 7997	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (2^nd ‘𝑝) = (𝑐 ↾ 𝑅))
325	324	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((2^nd ‘𝑝)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
326	325	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘((𝑐 ↾ 𝑅)‘𝑡)))
327	326	prodeq2ad 46235	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)))
328	323, 327	oveq12d 7429	. . . . . . . . 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 6886	. . . . . . . . . . 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 6884	. . . . . . . . . 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 46235	. . . . . . . . 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 7429	. . . . . . . 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 7427	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
334	333	fveq2d 6886	. . . . . . . . 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 6884	. . . . . . . 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 7429	. . . . . . 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 7429	. . . . . 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 7419	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑_m 𝑠) = ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})))
339		rabeq 3436	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 15740	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
342	341	eqeq1d 2771	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛))
343	342	rabbidv 3429	. . . . . . . . . . . 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 2804	. . . . . . . . . . 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 5207	. . . . . . . . . 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 4755	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ⊆ 𝑇)
347	4, 346	unssd 4151	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
348	3, 347	ssexd 5295	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
349		elpwg 4568	. . . . . . . . . . . 12 ⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
350	348, 349	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
351	347, 350	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
352	63	mptex 7222	. . . . . . . . . . 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 7014	. . . . . . . . 9 ⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
355		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
356	355	oveq1d 7426	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑_m (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
357		rabeq 3436	. . . . . . . . . . . 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 18	. . . . . . . . . . 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 2781	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
360	359	rabbidv 3429	. . . . . . . . . . 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 2804	. . . . . . . . . 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 486	. . . . . . . . 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 7444	. . . . . . . . . . 11 ⊢ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ V
364	363	rabex 5310	. . . . . . . . . 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 6998	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
367		fzfid 14009	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝐽) ∈ Fin)
368		snfi 9040	. . . . . . . . . . . 12 ⊢ {𝑍} ∈ Fin
369	368	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑍} ∈ Fin)
370		unfi 9155	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ Fin)
371	6, 369, 370	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ Fin)
372		mapfi 9305	. . . . . . . . . 10 ⊢ (((0...𝐽) ∈ Fin ∧ (𝑅 ∪ {𝑍}) ∈ Fin) → ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ Fin)
373	367, 371, 372	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∈ Fin)
374		ssrab2 4040	. . . . . . . . . 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 9157	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin)
378	366, 377	eqeltrd 2869	. . . . . . 7 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin)
379	378	adantr 485	. . . . . 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 46587	. . . . . . 7 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
382	381	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
383		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
384		opex 5446	. . . . . . . . 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 7002	. . . . . . . 8 ⊢ ((𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
387	383, 385, 386	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
388	387	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
389	45	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℕ₀)
390		eliun 4962	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
391	390	bilani 509	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
392		nfv 1941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝜑
393		nfcv 2931	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑝
394		nfiu1 4994	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
395	393, 394	nfel 2945	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
396	392, 395	nfan 1926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
397		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(1^st ‘𝑝) ∈ (0...𝐽)
398		xp1st 8018	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ {𝑘})
399		elsni 4609	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑝) ∈ {𝑘} → (1^st ‘𝑝) = 𝑘)
400	398, 399	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) = 𝑘)
401	400	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
402		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
403	401, 402	eqeltrd 2869	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
404	403	ex 417	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
405	404	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))))
406	396, 397, 405	rexlimd 3278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
407	391, 406	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
408		elfzelz 13552	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℤ)
409	407, 408	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
410	389, 409	bccld 45961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℕ₀)
411	410	nn0cnd 12567	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
412	411	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
413		elfznn0 13648	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℕ₀)
414	407, 413	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℕ₀)
415	414	faccld 14320	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℕ)
416	415	nncnd 12249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
417	416	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
418	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
419		nfv 1941	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(2^nd ‘𝑝):𝑅⟶(0...𝐽)
420	84, 82	eqsstrd 3977	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝑘) ↑_m 𝑅))
421		ovex 7444	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐽) ∈ V
422	421	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝐽) ∈ V)
423		mapss 8887	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝐽) ∈ V ∧ (0...𝑘) ⊆ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
424	422, 122, 423	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
425	424	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_m 𝑅) ⊆ ((0...𝐽) ↑_m 𝑅))
426	420, 425	sstrd 3953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_m 𝑅))
427	426	3adant3 1148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_m 𝑅))
428		xp2nd 8019	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
429	428	3ad2ant3 1151	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
430	427, 429	sseldd 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((0...𝐽) ↑_m 𝑅))
431		elmapi 8846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑝) ∈ ((0...𝐽) ↑_m 𝑅) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
432	430, 431	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
433	432	3exp 1135	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
434	433	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
435	396, 419, 434	rexlimd 3278	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽)))
436	391, 435	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
437	436	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽))
438		elfznn0 13648	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((2^nd ‘𝑝)‘𝑡) ∈ ℕ₀)
439	438	faccld 14320	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
440	439	nncnd 12249	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
441	437, 440	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
442	418, 441	fprodcl 16006	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
443	442	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
444	437, 439	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
445		nnne0 12270	. . . . . . . . . . . . 13 ⊢ ((!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
446	444, 445	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
447	418, 441, 446	fprodn0 16033	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
448	447	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
449	417, 443, 448	divcld 11991	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) ∈ ℂ)
450	7	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
451		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝜑)
452	451, 13	sylancom 599	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
453	451, 133	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
454	453, 437	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))
455	451, 452, 454	3jca 1144	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
456		eleq1 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
457	456	3anbi3d 1468	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))))
458		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)))
459	458	feq1d 6688	. . . . . . . . . . . . . . 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))
460	457, 459	imbi12d 347	. . . . . . . . . . . . . 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)))
461	460, 145	vtoclg 3529	. . . . . . . . . . . . 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))
462	437, 455, 461	sylc 66	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)
463	462	adantllr 731	. . . . . . . . . . 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)
464	17	adantlr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
465	463, 464	ffvelcdmd 7081	. . . . . . . . . 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)
466	450, 465	fprodcl 16006	. . . . . . . . 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)
467	449, 466	mulcld 11229	. . . . . . . 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)
468		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)
469		simp1 1152	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
470	403	3adant1 1146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
471		fznn0sub2 13663	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
472	471	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ (0...𝐽)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
473	469, 470, 472	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
474	473	3exp 1135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
475	474	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
476	396, 468, 475	rexlimd 3278	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)))
477	391, 476	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
478		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
479	478, 23	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇)
480	478, 133	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ⊆ (0...𝑁))
481	480, 477	sseldd 3944	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))
482	478, 479, 481	3jca 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
483		eleq1 2857	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → (𝑗 ∈ (0...𝑁) ↔ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
484	483	3anbi3d 1468	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))))
485		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))))
486	485	feq1d 6688	. . . . . . . . . . . . 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))
487	484, 486	imbi12d 347	. . . . . . . . . . . 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)))
488		simp2 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → 𝑍 ∈ 𝑇)
489		id 23	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)))
490	26	3anbi2d 1467	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁))))
491	179	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
492	491	feq1d 6688	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
493	490, 492	imbi12d 347	. . . . . . . . . . . . . 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)))
494	493, 145	vtoclg 3529	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
495	488, 489, 494	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
496	487, 495	vtoclg 3529	. . . . . . . . . . 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))
497	477, 482, 496	sylc 66	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
498	497	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
499	34	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑥 ∈ 𝑋)
500	498, 499	ffvelcdmd 7081	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) ∈ ℂ)
501	467, 500	mulcld 11229	. . . . . . 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)
502	412, 501	mulcld 11229	. . . . . 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)
503	337, 379, 382, 388, 502	fsumf1o 15774	. . . . 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))))
504		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝜑)
505	366	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
506	383, 505	eleqtrd 2871	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
507	374	sseli 3939	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} → 𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
508	506, 507	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})))
509		elmapi 8846	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0...𝐽) ↑_m (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
510	508, 509	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
511		snidg 4629	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍})
512	23, 511	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
513		elun2 4142	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
514	512, 513	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍}))
515	514	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
516	510, 515	ffvelcdmd 7081	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽))
517		0zd 12603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ∈ ℤ)
518	124	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℤ)
519		fzssz 13554	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℤ
520	519	sseli 3939	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℤ)
521	520	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℤ)
522	518, 521	zsubcld 12705	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
523		elfzle2 13556	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽)
524	523	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ≤ 𝐽)
525	163	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
526	521	zred 12700	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℝ)
527	525, 526	subge0d 11804	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽))
528	524, 527	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍)))
529		elfzle1 13555	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍))
530	529	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝑐‘𝑍))
531	525, 526	subge02d 11806	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))
532	530, 531	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)
533	517, 518, 522, 528, 532	elfzd 13543	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
534	504, 516, 533	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
535		bcval2 14341	. . . . . . . . . . 11 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
536	534, 535	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
537	163	recnd 11237	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℂ)
538	537	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
539		zsscn 12599	. . . . . . . . . . . . . . . . 17 ⊢ ℤ ⊆ ℂ
540	519, 539	sstri 3952	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℂ
541	540	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
542	541, 516	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ)
543	538, 542	nncand 11574	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍))
544	543	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = (!‘(𝑐‘𝑍)))
545	544	oveq1d 7426	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍)))))
546	545	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
547	45	faccld 14320	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘𝐽) ∈ ℕ)
548	547	nncnd 12249	. . . . . . . . . . . . 13 ⊢ (𝜑 → (!‘𝐽) ∈ ℂ)
549	548	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
550		elfznn0 13648	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℕ₀)
551	516, 550	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℕ₀)
552	551	faccld 14320	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℕ)
553	552	nncnd 12249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
554		elfznn0 13648	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
555	534, 554	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
556	555	faccld 14320	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℕ)
557	556	nncnd 12249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
558	552	nnne0d 12286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
559	556	nnne0d 12286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
560	549, 553, 557, 558, 559	divdiv1d 12022	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
561	560	eqcomd 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
562	536, 546, 561	3eqtrd 2808	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
563	562	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
564		fvres 6901	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
565	564	fveq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → (!‘((𝑐 ↾ 𝑅)‘𝑡)) = (!‘(𝑐‘𝑡)))
566	565	prodeq2i 15972	. . . . . . . . . . . . . . 15 ⊢ ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))
567	566	oveq2i 7422	. . . . . . . . . . . . . 14 ⊢ ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
568	564	fveq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
569	568	fveq1d 6884	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑅 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
570	569	prodeq2i 15972	. . . . . . . . . . . . . 14 ⊢ ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)
571	567, 570	oveq12i 7423	. . . . . . . . . . . . 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))
572	571	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)))
573	572	adantlr 727	. . . . . . . . . . 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)))
574	557	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
575	504, 6	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
576	75	ssriv 3947	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐽) ⊆ ℕ₀
577	576	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℕ₀)
578	510	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
579		elun1 4141	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝑅 → 𝑡 ∈ (𝑅 ∪ {𝑍}))
580	579	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
581	578, 580	ffvelcdmd 7081	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽))
582	577, 581	sseldd 3944	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
583	582	faccld 14320	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
584	583	nncnd 12249	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
585	575, 584	fprodcl 16006	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
586	585	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
587	7	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
588	504	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
589	504, 13	sylan 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
590	588, 133	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
591	590, 581	sseldd 3944	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
592	588, 589, 591, 146	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
593	592	adantllr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
594	17	adantlr 727	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
595	593, 594	ffvelcdmd 7081	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
596	587, 595	fprodcl 16006	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
597	575, 583	fprodnncl 16009	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ)
598		nnne0 12270	. . . . . . . . . . . . . 14 ⊢ (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
599	597, 598	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
600	599	adantlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
601	574, 586, 596, 600	div32d 12014	. . . . . . . . . . 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)))))
602	573, 601	eqtrd 2804	. . . . . . . . . 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)))))
603	543	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
604	603	fveq1d 6884	. . . . . . . . . . 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))
605	604	adantlr 727	. . . . . . . . . 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))
606	602, 605	oveq12d 7429	. . . . . . . . 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)))
607	596, 586, 600	divcld 11991	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
608	504, 23	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
609	504, 133	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ (0...𝑁))
610	609, 516	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝑁))
611	504, 608, 610	3jca 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)))
612		eleq1 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑍) ∈ (0...𝑁)))
613	612	3anbi3d 1468	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁))))
614		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
615	614	feq1d 6688	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
616	613, 615	imbi12d 347	. . . . . . . . . . . . . 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)))
617	616, 495	vtoclg 3529	. . . . . . . . . . . . 13 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
618	516, 611, 617	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
619	618	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
620	34	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑥 ∈ 𝑋)
621	619, 620	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥) ∈ ℂ)
622	574, 607, 621	mulassd 11232	. . . . . . . . 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))))
623	606, 622	eqtrd 2804	. . . . . . . 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))))
624	563, 623	oveq12d 7429	. . . . . . 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)))))
625	548	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
626	553	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
627	558	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
628	625, 626, 627	divcld 11991	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / (!‘(𝑐‘𝑍))) ∈ ℂ)
629	607, 621	mulcld 11229	. . . . . . . 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)
630	559	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
631	628, 574, 629, 630	dmmcand 45959	. . . . . . 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))))
632	596, 621, 586, 600	div23d 12028	. . . . . . . . . . 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)))
633	632	eqcomd 2775	. . . . . . . . . 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))))
634		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
635		nfcv 2931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)
636	608	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
637	11	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
638		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (𝑐‘𝑡) = (𝑐‘𝑍))
639	179, 638	fveq12d 6889	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
640	639	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))
641	634, 635, 587, 636, 637, 595, 640, 621	fprodsplitsn 16043	. . . . . . . . . . . 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)))
642	641	eqcomd 2775	. . . . . . . . . . 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))
643	642	oveq1d 7426	. . . . . . . . . 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))))
644	633, 643	eqtrd 2804	. . . . . . . . 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))))
645	644	oveq2d 7427	. . . . . . . 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)))))
646	587, 368, 370	sylancl 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
647	504	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝜑)
648	347	sselda 3943	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
649	648	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
650	510, 609	fssd 6724	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝑁))
651	650	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝑁))
652	647, 649, 651, 146	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
653	652	adantllr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
654	620	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑥 ∈ 𝑋)
655	653, 654	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
656	646, 655	fprodcl 16006	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
657	625, 626, 656, 586, 627, 600	divmuldivd 12032	. . . . . . . 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)))))
658	553, 585	mulcomd 11230	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
659		nfv 1941	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
660		nfcv 2931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(!‘(𝑐‘𝑍))
661	504, 10	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
662	638	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑍)))
663	659, 660, 575, 608, 661, 584, 662, 553	fprodsplitsn 16043	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
664	663	eqcomd 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
665	658, 664	eqtrd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
666	665	oveq2d 7427	. . . . . . . . . 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))))
667	666	adantlr 727	. . . . . . . . 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))))
668	504, 371	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
669	576	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0...𝐽) ⊆ ℕ₀)
670	510	ffvelcdmda 7080	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝐽))
671	669, 670	sseldd 3944	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ ℕ₀)
672	671	faccld 14320	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℕ)
673	672	nncnd 12249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℂ)
674	668, 673	fprodcl 16006	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
675	674	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
676	672	nnne0d 12286	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ≠ 0)
677	668, 673, 676	fprodn0 16033	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
678	677	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
679	625, 656, 675, 678	div23d 12028	. . . . . . . . 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)))
680		eqidd 2770	. . . . . . . . 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)))
681	667, 679, 680	3eqtrd 2808	. . . . . . . 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)))
682	645, 657, 681	3eqtrd 2808	. . . . . . 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)))
683	624, 631, 682	3eqtrd 2808	. . . . . 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)))
684	683	sumeq2dv 15753	. . . . 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)))
685	503, 684	eqtrd 2804	. . . 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)))
686	294, 318, 685	3eqtrd 2808	. . 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)))
687	686	mpteq2dva 5206	. 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))))
688	39, 209, 687	3eqtrd 2808	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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3422 Vcvv 3461 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 𝒫 cpw 4565 {csn 4592 {cpr 4594 ⟨cop 4598 ∪ ciun 4958 class class class wbr 5111 ↦ cmpt 5194 × cxp 5660 ↾ cres 5664 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 1^st c1st 7984 2^nd c2nd 7985 ↑_m cmap 8824 Fincfn 8943 ℂcc 11098 ℝcr 11099 0cc0 11100 · cmul 11105 ≤ cle 11244 − cmin 11441 / cdiv 11871 ℕcn 12233 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ...cfz 13535 !cfa 14309 Ccbc 14338 Σcsu 15737 ∏cprod 15957 ↾_t crest 17473 TopOpenctopn 17474 ℂ_fldccnfld 21491 D^𝑛 cdvn 25992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-prod 15958 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-tx 23688 df-hmeo 23881 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-limc 25994 df-dv 25995 df-dvn 25996

This theorem is referenced by: dvnprodlem3 46589

W3C validator