dvnprodlem2 - Mathbox for Glauco Siliprandi

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Theorem dvnprodlem2 38837

Description: Induction step for dvnprodlem2 38837. (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 1830	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ 𝑋)
2		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑡((𝐻‘𝑍)‘𝑥)
3		dvnprodlem2.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ Fin)
4		dvnprodlem2.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ⊆ 𝑇)
5		ssfi 8065	. . . . . . . 8 ⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin)
6	3, 4, 5	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Fin)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ Fin)
8		dvnprodlem2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ (𝑇 ∖ 𝑅))
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ (𝑇 ∖ 𝑅))
10	8	eldifbd 3553	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑍 ∈ 𝑅)
12		simpl 472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝜑)
13	4	sselda 3568	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
14		dvnprodlem2.h	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ)
15	12, 13, 14	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
16	15	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → (𝐻‘𝑡):𝑋⟶ℂ)
17		simplr 788	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
18	16, 17	ffvelrnd 6268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑅) → ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
19		fveq2 6103	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝐻‘𝑡) = (𝐻‘𝑍))
20	19	fveq1d 6105	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡)‘𝑥) = ((𝐻‘𝑍)‘𝑥))
21		id 22	. . . . . . . . 9 ⊢ (𝜑 → 𝜑)
22		eldifi 3694	. . . . . . . . . 10 ⊢ (𝑍 ∈ (𝑇 ∖ 𝑅) → 𝑍 ∈ 𝑇)
23	8, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
24		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → 𝑍 ∈ 𝑇)
25		id 22	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝜑 ∧ 𝑍 ∈ 𝑇))
26		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑇 ↔ 𝑍 ∈ 𝑇))
27	26	anbi2d 736	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇)))
28	19	feq1d 5943	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑍 → ((𝐻‘𝑡):𝑋⟶ℂ ↔ (𝐻‘𝑍):𝑋⟶ℂ))
29	27, 28	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑍 → (((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡):𝑋⟶ℂ) ↔ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)))
30	29, 14	vtoclg 3239	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ))
31	24, 25, 30	sylc 63	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇) → (𝐻‘𝑍):𝑋⟶ℂ)
32	21, 23, 31	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘𝑍):𝑋⟶ℂ)
33	32	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐻‘𝑍):𝑋⟶ℂ)
34		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
35	33, 34	ffvelrnd 6268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐻‘𝑍)‘𝑥) ∈ ℂ)
36	1, 2, 7, 9, 11, 18, 20, 35	fprodsplitsn 14559	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥) = (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥)))
37	36	mpteq2dva 4672	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ (𝑅 ∪ {𝑍})((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ (∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) · ((𝐻‘𝑍)‘𝑥))))
38	37	oveq2d 6565	. . 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 6105	. 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 14562	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥) ∈ ℂ)
43		dvnprodlem2.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (0...𝑁))
44		elfznn0 12302	. . . 4 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ∈ ℕ₀)
45	43, 44	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
46		eqid 2610	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
47		eqid 2610	. . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))
48		dvnprodlem2.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}))
49	48	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})))
50		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 𝑅))
51		rabeq 3166	. . . . . . . . . . . . . 14 ⊢ (((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})
52	50, 51	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
53		sumeq1 14267	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡))
54	53	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛))
55	54	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
56	52, 55	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})
57	56	mpteq2dv 4673	. . . . . . . . . . 11 ⊢ (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}))
58	57	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 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}))
59		ssexg 4732	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → 𝑅 ∈ V)
60	4, 3, 59	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ V)
61		elpwg 4116	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
62	60, 61	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇))
63	4, 62	mpbird 246	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇)
64	63	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑅 ∈ 𝒫 𝑇)
65		nn0ex 11175	. . . . . . . . . . . 12 ⊢ ℕ₀ ∈ V
66	65	mptex 6390	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V
67	66	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V)
68	49, 58, 64, 67	fvmptd 6197	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}))
69		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘))
70	69	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑_𝑚 𝑅) = ((0...𝑘) ↑_𝑚 𝑅))
71		rabeq 3166	. . . . . . . . . . . 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})
72	70, 71	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})
73		eqeq2 2621	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘))
74	73	rabbidv 3164	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
75	72, 74	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
76	75	adantl 481	. . . . . . . . 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})
77		elfznn0 12302	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ₀)
78	77	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ₀)
79		fzfid 12634	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...𝑘) ∈ Fin)
80		mapfi 8145	. . . . . . . . . . . 12 ⊢ (((0...𝑘) ∈ Fin ∧ 𝑅 ∈ Fin) → ((0...𝑘) ↑_𝑚 𝑅) ∈ Fin)
81	79, 6, 80	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → ((0...𝑘) ↑_𝑚 𝑅) ∈ Fin)
82	81	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_𝑚 𝑅) ∈ Fin)
83		ssrab2 3650	. . . . . . . . . . 11 ⊢ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑_𝑚 𝑅)
84	83	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ⊆ ((0...𝑘) ↑_𝑚 𝑅))
85	82, 84	ssexd 4733	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V)
86	68, 76, 78, 85	fvmptd 6197	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
87		ssfi 8065	. . . . . . . . . 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)
88	81, 83, 87	sylancl 693	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
89	88	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ Fin)
90	86, 89	eqeltrd 2688	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
91	90	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
92	77	faccld 12933	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℕ)
93	92	nncnd 10913	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → (!‘𝑘) ∈ ℂ)
94	93	ad2antlr 759	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (!‘𝑘) ∈ ℂ)
95	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
96	95	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
97		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
98	97	ssriv 3572	. . . . . . . . . . . . . 14 ⊢ (0...𝑘) ⊆ ℕ₀
99	98	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℕ₀)
100		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))
101	86	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}))
102	101	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (𝑐 ∈ ((𝐶‘𝑅)‘𝑘) ↔ 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}))
103	100, 102	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘})
104	83	sseli 3564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → 𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅))
105	103, 104	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅))
106		elmapi 7765	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ((0...𝑘) ↑_𝑚 𝑅) → 𝑐:𝑅⟶(0...𝑘))
107	105, 106	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑐:𝑅⟶(0...𝑘))
108	107	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑐:𝑅⟶(0...𝑘))
109		simpr 476	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅)
110	108, 109	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑘))
111	99, 110	sseldd 3569	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
112	111	faccld 12933	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
113	112	nncnd 10913	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
114	96, 113	fprodcl 14521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
115	112	nnne0d 10942	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ≠ 0)
116	96, 113, 115	fprodn0 14548	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
117	94, 114, 116	divcld 10680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
118	117	adantlr 747	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
119	96	adantlr 747	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → 𝑅 ∈ Fin)
120	21	ad4antr 764	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
121	120, 13	sylancom 698	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
122		elfzuz3 12210	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ (ℤ_≥‘𝑘))
123		fzss2 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝐽 ∈ (ℤ_≥‘𝑘) → (0...𝑘) ⊆ (0...𝐽))
124	122, 123	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝑘) ⊆ (0...𝐽))
125	124	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝐽))
126	45	nn0zd 11356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ∈ ℤ)
127		dvnprodlem2.n	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
128	127	nn0zd 11356	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
129		elfzle2 12216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐽 ∈ (0...𝑁) → 𝐽 ≤ 𝑁)
130	43, 129	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
131	126, 128, 130	3jca 1235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
132		eluz2 11569	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) ↔ (𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐽 ≤ 𝑁))
133	131, 132	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝐽))
134		fzss2 12252	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ_≥‘𝐽) → (0...𝐽) ⊆ (0...𝑁))
135	133, 134	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝐽) ⊆ (0...𝑁))
136	135	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝐽) ⊆ (0...𝑁))
137	125, 136	sstrd 3578	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0...𝑘) ⊆ (0...𝑁))
138	137	ad2antrr 758	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ (0...𝑁))
139	138, 110	sseldd 3569	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
140	139	adantllr 751	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
141		fvex 6113	. . . . . . . . . . 11 ⊢ (𝑐‘𝑡) ∈ V
142		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑡) ∈ (0...𝑁)))
143	142	3anbi3d 1397	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁))))
144		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑐‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
145	144	feq1d 5943	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑐‘𝑡) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ))
146	143, 145	imbi12d 333	. . . . . . . . . . 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)))
147		dvnprodlem2.dvnh	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ)
148	141, 146, 147	vtocl 3232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ (𝑐‘𝑡) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
149	120, 121, 140, 148	syl3anc 1318	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
150		simpllr 795	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
151	149, 150	ffvelrnd 6268	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
152	119, 151	fprodcl 14521	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
153	118, 152	mulcld 9939	. . . . . 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)
154	91, 153	fsumcl 14311	. . . . 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)
155		eqid 2610	. . . . 5 ⊢ (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)))
156	154, 155	fmptd 6292	. . . 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)
157		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))))
158	157	adantr 480	. . . . . 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))))
159		0zd 11266	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ∈ ℤ)
160	128	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℤ)
161		elfzelz 12213	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ)
162	161	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℤ)
163	159, 160, 162	3jca 1235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
164		elfzle1 12215	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ 𝑘)
165	164	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘)
166	162	zred 11358	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ)
167	45	nn0red 11229	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ ℝ)
168	167	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
169	160	zred 11358	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑁 ∈ ℝ)
170		elfzle2 12216	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽)
171	170	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽)
172	130	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ≤ 𝑁)
173	166, 168, 169, 171, 172	letrd 10073	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝑁)
174	163, 165, 173	jca32 556	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
175		elfz2 12204	. . . . . . 7 ⊢ (𝑘 ∈ (0...𝑁) ↔ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (0 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
176	174, 175	sylibr 223	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ (0...𝑁))
177		rspa 2914	. . . . . 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))))
178	158, 176, 177	syl2anc 691	. . . . 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))))
179	178	feq1d 5943	. . . 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))
180	156, 179	mpbird 246	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘):𝑋⟶ℂ)
181	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑍 ∈ 𝑇)
182		simpl 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝜑)
183	182, 181, 176	3jca 1235	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)))
184	26	3anbi2d 1396	. . . . . . 7 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
185	19	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑡 = 𝑍 → (𝑆 D^𝑛 (𝐻‘𝑡)) = (𝑆 D^𝑛 (𝐻‘𝑍)))
186	185	fveq1d 6105	. . . . . . . 8 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
187	186	feq1d 5943	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
188	184, 187	imbi12d 333	. . . . . 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)))
189		eleq1 2676	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...𝑁)))
190	189	3anbi3d 1397	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁))))
191		fveq2 6103	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘))
192	191	feq1d 5943	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ))
193	190, 192	imbi12d 333	. . . . . . 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)))
194	193, 147	chvarv 2251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑘):𝑋⟶ℂ)
195	188, 194	vtoclg 3239	. . . . 5 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
196	181, 183, 195	sylc 63	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ)
197	32	feqmptd 6159	. . . . . . . . 9 ⊢ (𝜑 → (𝐻‘𝑍) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
198	197	eqcomd 2616	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)) = (𝐻‘𝑍))
199	198	oveq2d 6565	. . . . . . 7 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
200	199	fveq1d 6105	. . . . . 6 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
201	200	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘))
202	201	feq1d 5943	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ))
203	196, 202	mpbird 246	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘):𝑋⟶ℂ)
204		fveq2 6103	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑡)‘𝑦) = ((𝐻‘𝑡)‘𝑥))
205	204	prodeq2ad 38659	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦) = ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
206	205	cbvmptv 4678	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥))
207	206	oveq2i 6560	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))
208	207	fveq1i 6104	. . . 4 ⊢ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ∏𝑡 ∈ 𝑅 ((𝐻‘𝑡)‘𝑥)))‘𝑘)
209	208	mpteq2i 4669	. . 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))
210		fveq2 6103	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐻‘𝑍)‘𝑦) = ((𝐻‘𝑍)‘𝑥))
211	210	cbvmptv 4678	. . . . . 6 ⊢ (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)) = (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥))
212	211	oveq2i 6560	. . . . 5 ⊢ (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))
213	212	fveq1i 6104	. . . 4 ⊢ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑥)))‘𝑘)
214	213	mpteq2i 4669	. . 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))
215	40, 41, 42, 35, 45, 46, 47, 180, 203, 209, 214	dvnmul 38833	. 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)))))
216	208	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))
217	157	r19.21bi 2916	. . . . . . . . . . . . . . 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))))
218	182, 176, 217	syl2anc 691	. . . . . . . . . . . . . 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))))
219	216, 218	eqtrd 2644	. . . . . . . . . . . . 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))))
220	219	mpteq2dva 4672	. . . . . . . . . . . 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)))))
221		mptexg 6389	. . . . . . . . . . . . . 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)
222	41, 221	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)
223	222	adantr 480	. . . . . . . . . . . 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)
224	220, 223	fvmpt2d 6202	. . . . . . . . . . 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))))
225	224	adantlr 747	. . . . . . . . . 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))))
226	225	fveq1d 6105	. . . . . . . . 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))
227	34	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → 𝑥 ∈ 𝑋)
228	154	an32s 842	. . . . . . . . . 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)
229	155	fvmpt2 6200	. . . . . . . . . 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)))
230	227, 228, 229	syl2anc 691	. . . . . . . . 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)))
231	226, 230	eqtrd 2644	. . . . . . . 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)))
232		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘) = ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗))
233	232	cbvmptv 4678	. . . . . . . . . . . . . 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))
234	233	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)))
235	212, 199	syl5eq 2656	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦))) = (𝑆 D^𝑛 (𝐻‘𝑍)))
236	235	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
237	236	mpteq2dv 4673	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑗)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗)))
238	234, 237	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝑦 ∈ 𝑋 ↦ ((𝐻‘𝑍)‘𝑦)))‘𝑘)) = (𝑗 ∈ (0...𝐽) ↦ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗)))
239	238	adantr 480	. . . . . . . . . . 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)))
240		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
241	240	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑗 = (𝐽 − 𝑘)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
242		0zd 11266	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 0 ∈ ℤ)
243		elfzel2 12211	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ)
244	243, 161	zsubcld 11363	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ∈ ℤ)
245	242, 243, 244	3jca 1235	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ))
246	243	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ)
247	77	nn0red 11229	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ)
248	246, 247	subge0d 10496	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽))
249	170, 248	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘))
250	246, 247	subge02d 10498	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽))
251	164, 250	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (𝐽 − 𝑘) ≤ 𝐽)
252	245, 249, 251	jca32 556	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...𝐽) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
253	252	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
254		elfz2 12204	. . . . . . . . . . . 12 ⊢ ((𝐽 − 𝑘) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − 𝑘) ∈ ℤ) ∧ (0 ≤ (𝐽 − 𝑘) ∧ (𝐽 − 𝑘) ≤ 𝐽)))
255	253, 254	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ∈ (0...𝐽))
256		fvex 6113	. . . . . . . . . . . 12 ⊢ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V
257	256	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)) ∈ V)
258	239, 241, 255, 257	fvmptd 6197	. . . . . . . . . 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)))
259	258	adantlr 747	. . . . . . . . 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)))
260	259	fveq1d 6105	. . . . . . . 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))
261	231, 260	oveq12d 6567	. . . . . . 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)))
262	261	oveq2d 6565	. . . . . 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))))
263	90	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ∈ Fin)
264		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝐽 − 𝑘) ∈ V
265		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − 𝑘) → (𝑗 ∈ (0...𝐽) ↔ (𝐽 − 𝑘) ∈ (0...𝐽)))
266	265	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − 𝑘) → ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽))))
267	240	feq1d 5943	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − 𝑘) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ))
268	266, 267	imbi12d 333	. . . . . . . . . . . 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)))
269		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (0...𝐽) ↔ 𝑗 ∈ (0...𝐽)))
270	269	anbi2d 736	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ 𝑗 ∈ (0...𝐽))))
271		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
272	271	feq1d 5943	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑗 → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑘):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
273	270, 272	imbi12d 333	. . . . . . . . . . . . 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)))
274	273, 196	chvarv 2251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
275	264, 268, 274	vtocl 3232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐽 − 𝑘) ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
276	182, 255, 275	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
277	276	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)):𝑋⟶ℂ)
278	277, 227	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
279		anass 679	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)))
280		ancom 465	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽)))
281	280	anbi2i 726	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))))
282		anass 679	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ↔ (𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))))
283	282	bicomi 213	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑘 ∈ (0...𝐽))) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
284	281, 283	bitri 263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑥 ∈ 𝑋)) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
285	279, 284	bitri 263	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)))
286	285	anbi1i 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)))
287	286	imbi1i 338	. . . . . . . . 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))
288	153, 287	mpbi 219	. . . . . . . 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)
289	263, 278, 288	fsummulc1 14359	. . . . . . 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)))
290	289	oveq2d 6565	. . . . . 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))))
291	182, 45	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℕ₀)
292	291, 162	bccld 38472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℕ₀)
293	292	nn0cnd 11230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
294	293	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) → (𝐽C𝑘) ∈ ℂ)
295	278	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥) ∈ ℂ)
296	288, 295	mulcld 9939	. . . . . . 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)
297	263, 294, 296	fsummulc2 14358	. . . . . 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))))
298	262, 290, 297	3eqtrd 2648	. . . . 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))))
299	298	sumeq2dv 14281	. . . 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))))
300		vex 3176	. . . . . . . 8 ⊢ 𝑘 ∈ V
301		vex 3176	. . . . . . . 8 ⊢ 𝑐 ∈ V
302	300, 301	op1std 7069	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (1^st ‘𝑝) = 𝑘)
303	302	oveq2d 6565	. . . . . 6 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C𝑘))
304	302	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘(1^st ‘𝑝)) = (!‘𝑘))
305	300, 301	op2ndd 7070	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (2^nd ‘𝑝) = 𝑐)
306	305	fveq1d 6105	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((2^nd ‘𝑝)‘𝑡) = (𝑐‘𝑡))
307	306	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘(𝑐‘𝑡)))
308	307	prodeq2ad 38659	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
309	304, 308	oveq12d 6567	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) = ((!‘𝑘) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))))
310	306	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
311	310	fveq1d 6105	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
312	311	prodeq2ad 38659	. . . . . . . 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))
313	309, 312	oveq12d 6567	. . . . . . 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)))
314	302	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − 𝑘))
315	314	fveq2d 6107	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘)))
316	315	fveq1d 6105	. . . . . . 7 ⊢ (𝑝 = ⟨𝑘, 𝑐⟩ → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − 𝑘))‘𝑥))
317	313, 316	oveq12d 6567	. . . . . 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)))
318	303, 317	oveq12d 6567	. . . . 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))))
319		fzfid 12634	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (0...𝐽) ∈ Fin)
320	294	adantrr 749	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑐 ∈ ((𝐶‘𝑅)‘𝑘))) → (𝐽C𝑘) ∈ ℂ)
321	296	anasss 677	. . . . . 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)
322	320, 321	mulcld 9939	. . . . 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)
323	318, 319, 263, 322	fsum2d 14344	. . . 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))))
324		ovex 6577	. . . . . . . . 9 ⊢ (𝐽 − (𝑐‘𝑍)) ∈ V
325	301	resex 5363	. . . . . . . . 9 ⊢ (𝑐 ↾ 𝑅) ∈ V
326	324, 325	op1std 7069	. . . . . . . 8 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (1^st ‘𝑝) = (𝐽 − (𝑐‘𝑍)))
327	326	oveq2d 6565	. . . . . . 7 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽C(1^st ‘𝑝)) = (𝐽C(𝐽 − (𝑐‘𝑍))))
328	326	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘(1^st ‘𝑝)) = (!‘(𝐽 − (𝑐‘𝑍))))
329	324, 325	op2ndd 7070	. . . . . . . . . . . . 13 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (2^nd ‘𝑝) = (𝑐 ↾ 𝑅))
330	329	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((2^nd ‘𝑝)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡))
331	330	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (!‘((2^nd ‘𝑝)‘𝑡)) = (!‘((𝑐 ↾ 𝑅)‘𝑡)))
332	331	prodeq2ad 38659	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)))
333	328, 332	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))))
334	330	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)))
335	334	fveq1d 6105	. . . . . . . . . 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))
336	335	prodeq2ad 38659	. . . . . . . . 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))
337	333, 336	oveq12d 6567	. . . . . . . 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)))
338	326	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑝 = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ → (𝐽 − (1^st ‘𝑝)) = (𝐽 − (𝐽 − (𝑐‘𝑍))))
339	338	fveq2d 6107	. . . . . . . . 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)))))
340	339	fveq1d 6105	. . . . . . . 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))
341	337, 340	oveq12d 6567	. . . . . . 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)))
342	327, 341	oveq12d 6567	. . . . . 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))))
343	48	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})))
344		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑_𝑚 𝑠) = ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})))
345		rabeq 3166	. . . . . . . . . . . . . 14 ⊢ (((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})
346	344, 345	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})
347		sumeq1 14267	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡))
348	347	eqeq1d 2612	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛))
349	348	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (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})
350	346, 349	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (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})
351	350	mpteq2dv 4673	. . . . . . . . . . 11 ⊢ (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}))
352	351	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 = (𝑅 ∪ {𝑍})) → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
353	23	snssd 4281	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ⊆ 𝑇)
354	4, 353	unssd 3751	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇)
355	3, 354	ssexd 4733	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V)
356		elpwg 4116	. . . . . . . . . . . 12 ⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
357	355, 356	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇))
358	354, 357	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇)
359	65	mptex 6390	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V
360	359	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V)
361	343, 352, 358, 360	fvmptd 6197	. . . . . . . . 9 ⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}))
362		oveq2 6557	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽))
363	362	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑_𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
364		rabeq 3166	. . . . . . . . . . . 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})
365	363, 364	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})
366		eqeq2 2621	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽))
367	366	rabbidv 3164	. . . . . . . . . . 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})
368	365, 367	eqtrd 2644	. . . . . . . . . 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})
369	368	adantl 481	. . . . . . . . 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})
370		ovex 6577	. . . . . . . . . . 11 ⊢ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∈ V
371	370	rabex 4740	. . . . . . . . . 10 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V
372	371	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V)
373	361, 369, 45, 372	fvmptd 6197	. . . . . . . 8 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
374		fzfid 12634	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝐽) ∈ Fin)
375		snfi 7923	. . . . . . . . . . . 12 ⊢ {𝑍} ∈ Fin
376	375	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑍} ∈ Fin)
377		unfi 8112	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ Fin)
378	6, 376, 377	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ Fin)
379		mapfi 8145	. . . . . . . . . 10 ⊢ (((0...𝐽) ∈ Fin ∧ (𝑅 ∪ {𝑍}) ∈ Fin) → ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∈ Fin)
380	374, 378, 379	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∈ Fin)
381		ssrab2 3650	. . . . . . . . . 10 ⊢ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍}))
382	381	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
383		ssfi 8065	. . . . . . . . 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)
384	380, 382, 383	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ Fin)
385	373, 384	eqeltrd 2688	. . . . . . 7 ⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin)
386	385	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∈ Fin)
387		dvnprodlem2.d	. . . . . . . 8 ⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
388	48, 45, 387, 3, 23, 10, 354	dvnprodlem1 38836	. . . . . . 7 ⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
389	388	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
390		simpr 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
391		opex 4859	. . . . . . . . 9 ⊢ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V
392	391	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V)
393	387	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩ ∈ V) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
394	390, 392, 393	syl2anc 691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
395	394	adantlr 747	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = ⟨(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)⟩)
396	45	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℕ₀)
397		eliun 4460	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
398	397	biimpi 205	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
399	398	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))
400		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝜑
401		nfcv 2751	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑝
402		nfiu1 4486	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
403	401, 402	nfel 2763	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))
404	400, 403	nfan 1816	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))
405		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(1^st ‘𝑝) ∈ (0...𝐽)
406		xp1st 7089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ {𝑘})
407		elsni 4142	. . . . . . . . . . . . . . . . . 18 ⊢ ((1^st ‘𝑝) ∈ {𝑘} → (1^st ‘𝑝) = 𝑘)
408	406, 407	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) = 𝑘)
409	408	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) = 𝑘)
410		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽))
411	409, 410	eqeltrd 2688	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
412	411	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
413	412	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))))
414	404, 405, 413	rexlimd 3008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1^st ‘𝑝) ∈ (0...𝐽)))
415	399, 414	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
416		elfzelz 12213	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℤ)
417	415, 416	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℤ)
418	396, 417	bccld 38472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℕ₀)
419	418	nn0cnd 11230	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
420	419	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽C(1^st ‘𝑝)) ∈ ℂ)
421		elfznn0 12302	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (1^st ‘𝑝) ∈ ℕ₀)
422	415, 421	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ ℕ₀)
423	422	faccld 12933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℕ)
424	423	nncnd 10913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
425	424	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (!‘(1^st ‘𝑝)) ∈ ℂ)
426	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
427		nfv 1830	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(2^nd ‘𝑝):𝑅⟶(0...𝐽)
428	86, 84	eqsstrd 3602	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝑘) ↑_𝑚 𝑅))
429		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0...𝐽) ∈ V
430	429	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (0...𝐽) → (0...𝐽) ∈ V)
431		mapss 7786	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((0...𝐽) ∈ V ∧ (0...𝑘) ⊆ (0...𝐽)) → ((0...𝑘) ↑_𝑚 𝑅) ⊆ ((0...𝐽) ↑_𝑚 𝑅))
432	430, 124, 431	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0...𝐽) → ((0...𝑘) ↑_𝑚 𝑅) ⊆ ((0...𝐽) ↑_𝑚 𝑅))
433	432	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((0...𝑘) ↑_𝑚 𝑅) ⊆ ((0...𝐽) ↑_𝑚 𝑅))
434	428, 433	sstrd 3578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_𝑚 𝑅))
435	434	3adant3 1074	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) ⊆ ((0...𝐽) ↑_𝑚 𝑅))
436		xp2nd 7090	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
437	436	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘))
438	435, 437	sseldd 3569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝) ∈ ((0...𝐽) ↑_𝑚 𝑅))
439		elmapi 7765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘𝑝) ∈ ((0...𝐽) ↑_𝑚 𝑅) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
440	438, 439	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
441	440	3exp 1256	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
442	441	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))))
443	404, 427, 442	rexlimd 3008	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2^nd ‘𝑝):𝑅⟶(0...𝐽)))
444	399, 443	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2^nd ‘𝑝):𝑅⟶(0...𝐽))
445	444	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽))
446		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → ((2^nd ‘𝑝)‘𝑡) ∈ ℕ₀)
447	446	faccld 12933	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
448	447	nncnd 10913	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑝)‘𝑡) ∈ (0...𝐽) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
449	445, 448	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
450	426, 449	fprodcl 14521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
451	450	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℂ)
452	445, 447	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ)
453		nnne0 10930	. . . . . . . . . . . . 13 ⊢ ((!‘((2^nd ‘𝑝)‘𝑡)) ∈ ℕ → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
454	452, 453	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
455	426, 449, 454	fprodn0 14548	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
456	455	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡)) ≠ 0)
457	425, 451, 456	divcld 10680	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((!‘(1^st ‘𝑝)) / ∏𝑡 ∈ 𝑅 (!‘((2^nd ‘𝑝)‘𝑡))) ∈ ℂ)
458	7	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin)
459		simpll 786	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝜑)
460	459, 13	sylancom 698	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
461	459, 135	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
462	461, 445	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))
463	459, 460, 462	3jca 1235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
464		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → (𝑗 ∈ (0...𝑁) ↔ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁)))
465	464	3anbi3d 1397	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑡 ∈ 𝑇 ∧ ((2^nd ‘𝑝)‘𝑡) ∈ (0...𝑁))))
466		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = ((2^nd ‘𝑝)‘𝑡) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)))
467	466	feq1d 5943	. . . . . . . . . . . . . . 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))
468	465, 467	imbi12d 333	. . . . . . . . . . . . . 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)))
469	468, 147	vtoclg 3239	. . . . . . . . . . . . 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))
470	445, 463, 469	sylc 63	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((2^nd ‘𝑝)‘𝑡)):𝑋⟶ℂ)
471	470	adantllr 751	. . . . . . . . . . 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)
472	17	adantlr 747	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
473	471, 472	ffvelrnd 6268	. . . . . . . . . 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)
474	458, 473	fprodcl 14521	. . . . . . . . 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)
475	457, 474	mulcld 9939	. . . . . . . 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)
476		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)
477		simp1 1054	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
478	411	3adant1 1072	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1^st ‘𝑝) ∈ (0...𝐽))
479		fznn0sub2 12315	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑝) ∈ (0...𝐽) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
480	479	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1^st ‘𝑝) ∈ (0...𝐽)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
481	477, 478, 480	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
482	481	3exp 1256	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
483	482	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))))
484	404, 476, 483	rexlimd 3008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽)))
485	399, 484	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝐽))
486		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑)
487	486, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇)
488	486, 135	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ⊆ (0...𝑁))
489	488, 485	sseldd 3569	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))
490	486, 487, 489	3jca 1235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
491		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → (𝑗 ∈ (0...𝑁) ↔ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁)))
492	491	3anbi3d 1397	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝐽 − (1^st ‘𝑝)) ∈ (0...𝑁))))
493		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐽 − (1^st ‘𝑝)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))))
494	493	feq1d 5943	. . . . . . . . . . . . 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))
495	492, 494	imbi12d 333	. . . . . . . . . . . 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)))
496		simp2 1055	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → 𝑍 ∈ 𝑇)
497		id 22	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)))
498	26	3anbi2d 1396	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → ((𝜑 ∧ 𝑡 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁))))
499	185	fveq1d 6105	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗))
500	499	feq1d 5943	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
501	498, 500	imbi12d 333	. . . . . . . . . . . . . 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)))
502	501, 147	vtoclg 3239	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝑇 → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ))
503	496, 497, 502	sylc 63	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ)
504	495, 503	vtoclg 3239	. . . . . . . . . . 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))
505	485, 490, 504	sylc 63	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
506	505	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝))):𝑋⟶ℂ)
507	34	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑥 ∈ 𝑋)
508	506, 507	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (1^st ‘𝑝)))‘𝑥) ∈ ℂ)
509	475, 508	mulcld 9939	. . . . . . 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)
510	420, 509	mulcld 9939	. . . . . 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)
511	342, 386, 389, 395, 510	fsumf1o 14301	. . . . 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))))
512		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝜑)
513	373	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
514	390, 513	eleqtrd 2690	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})
515	381	sseli 3564	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} → 𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
516	514, 515	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})))
517		elmapi 7765	. . . . . . . . . . . . . 14 ⊢ (𝑐 ∈ ((0...𝐽) ↑_𝑚 (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
518	516, 517	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
519		snidg 4153	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍})
520	23, 519	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑍 ∈ {𝑍})
521		elun2 3743	. . . . . . . . . . . . . . 15 ⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍}))
522	520, 521	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍}))
523	522	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍}))
524	518, 523	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽))
525		0zd 11266	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ∈ ℤ)
526	126	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℤ)
527		fzssz 12214	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐽) ⊆ ℤ
528	527	sseli 3564	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℤ)
529	528	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℤ)
530	526, 529	zsubcld 11363	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ)
531	525, 526, 530	3jca 1235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ))
532		elfzle2 12216	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽)
533	532	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ≤ 𝐽)
534	167	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 𝐽 ∈ ℝ)
535	529	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝑐‘𝑍) ∈ ℝ)
536	534, 535	subge0d 10496	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽))
537	533, 536	mpbird 246	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍)))
538		elfzle1 12215	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍))
539	538	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → 0 ≤ (𝑐‘𝑍))
540	534, 535	subge02d 10498	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))
541	539, 540	mpbid 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)
542	531, 537, 541	jca32 556	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
543		elfz2 12204	. . . . . . . . . . . . 13 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)))
544	542, 543	sylibr 223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑐‘𝑍) ∈ (0...𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
545	512, 524, 544	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))
546		bcval2 12954	. . . . . . . . . . 11 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
547	545, 546	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))))
548	167	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐽 ∈ ℂ)
549	548	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ)
550		zsscn 11262	. . . . . . . . . . . . . . . . 17 ⊢ ℤ ⊆ ℂ
551	527, 550	sstri 3577	. . . . . . . . . . . . . . . 16 ⊢ (0...𝐽) ⊆ ℂ
552	551	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ)
553	552, 524	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ)
554	549, 553	nncand 10276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍))
555	554	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = (!‘(𝑐‘𝑍)))
556	555	oveq1d 6564	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍)))))
557	556	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) · (!‘(𝐽 − (𝑐‘𝑍))))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
558	45	faccld 12933	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘𝐽) ∈ ℕ)
559	558	nncnd 10913	. . . . . . . . . . . . 13 ⊢ (𝜑 → (!‘𝐽) ∈ ℂ)
560	559	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
561		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ∈ ℕ₀)
562	524, 561	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℕ₀)
563	562	faccld 12933	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℕ)
564	563	nncnd 10913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
565		elfznn0 12302	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
566	545, 565	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℕ₀)
567	566	faccld 12933	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℕ)
568	567	nncnd 10913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
569	563	nnne0d 10942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
570	567	nnne0d 10942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
571	560, 564, 568, 569, 570	divdiv1d 10711	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))) = ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))))
572	571	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / ((!‘(𝑐‘𝑍)) · (!‘(𝐽 − (𝑐‘𝑍))))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
573	547, 557, 572	3eqtrd 2648	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
574	573	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽C(𝐽 − (𝑐‘𝑍))) = (((!‘𝐽) / (!‘(𝑐‘𝑍))) / (!‘(𝐽 − (𝑐‘𝑍)))))
575		fvres 6117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))
576	575	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → (!‘((𝑐 ↾ 𝑅)‘𝑡)) = (!‘(𝑐‘𝑡)))
577	576	prodeq2i 14488	. . . . . . . . . . . . . . 15 ⊢ ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡)) = ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))
578	577	oveq2i 6560	. . . . . . . . . . . . . 14 ⊢ ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘((𝑐 ↾ 𝑅)‘𝑡))) = ((!‘(𝐽 − (𝑐‘𝑍))) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)))
579	575	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ 𝑅 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)))
580	579	fveq1d 6105	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ 𝑅 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥))
581	580	prodeq2i 14488	. . . . . . . . . . . . . 14 ⊢ ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘((𝑐 ↾ 𝑅)‘𝑡))‘𝑥) = ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥)
582	578, 581	oveq12i 6561	. . . . . . . . . . . . 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))
583	582	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)))
584	583	adantlr 747	. . . . . . . . . . 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)))
585	568	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ∈ ℂ)
586	512, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
587	77	ssriv 3572	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐽) ⊆ ℕ₀
588	587	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℕ₀)
589	518	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽))
590		elun1 3742	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ 𝑅 → 𝑡 ∈ (𝑅 ∪ {𝑍}))
591	590	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍}))
592	589, 591	ffvelrnd 6268	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽))
593	588, 592	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℕ₀)
594	593	faccld 12933	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℕ)
595	594	nncnd 10913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (!‘(𝑐‘𝑡)) ∈ ℂ)
596	586, 595	fprodcl 14521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
597	596	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℂ)
598	7	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin)
599	512	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝜑)
600	512, 13	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑇)
601	599, 135	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ (0...𝑁))
602	601, 592	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝑁))
603	599, 600, 602, 148	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
604	603	adantllr 751	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
605	17	adantlr 747	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑥 ∈ 𝑋)
606	604, 605	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
607	598, 606	fprodcl 14521	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
608	586, 594	fprodnncl 14524	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ)
609		nnne0 10930	. . . . . . . . . . . . . 14 ⊢ (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ∈ ℕ → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
610	608, 609	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
611	610	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) ≠ 0)
612	585, 597, 607, 611	div32d 10703	. . . . . . . . . . 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)))))
613	584, 612	eqtrd 2644	. . . . . . . . . 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)))))
614	554	fveq2d 6107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝐽 − (𝐽 − (𝑐‘𝑍)))) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
615	614	fveq1d 6105	. . . . . . . . . . 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))
616	615	adantlr 747	. . . . . . . . . 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))
617	613, 616	oveq12d 6567	. . . . . . . . 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)))
618	607, 597, 611	divcld 10680	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) / ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) ∈ ℂ)
619	512, 23	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
620	512, 135	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ (0...𝑁))
621	620, 524	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝑁))
622	512, 619, 621	3jca 1235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)))
623		eleq1 2676	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → (𝑗 ∈ (0...𝑁) ↔ (𝑐‘𝑍) ∈ (0...𝑁)))
624	623	3anbi3d 1397	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ 𝑗 ∈ (0...𝑁)) ↔ (𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁))))
625		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑐‘𝑍) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
626	625	feq1d 5943	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑐‘𝑍) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘𝑗):𝑋⟶ℂ ↔ ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
627	624, 626	imbi12d 333	. . . . . . . . . . . . . 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)))
628	627, 503	vtoclg 3239	. . . . . . . . . . . . 13 ⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → ((𝜑 ∧ 𝑍 ∈ 𝑇 ∧ (𝑐‘𝑍) ∈ (0...𝑁)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ))
629	524, 622, 628	sylc 63	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
630	629	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)):𝑋⟶ℂ)
631	34	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑥 ∈ 𝑋)
632	630, 631	ffvelrnd 6268	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥) ∈ ℂ)
633	585, 618, 632	mulassd 9942	. . . . . . . . 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))))
634	617, 633	eqtrd 2644	. . . . . . . 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))))
635	574, 634	oveq12d 6567	. . . . . . 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)))))
636	559	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘𝐽) ∈ ℂ)
637	564	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ∈ ℂ)
638	569	adantlr 747	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝑐‘𝑍)) ≠ 0)
639	636, 637, 638	divcld 10680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘𝐽) / (!‘(𝑐‘𝑍))) ∈ ℂ)
640	618, 632	mulcld 9939	. . . . . . . 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)
641	570	adantlr 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (!‘(𝐽 − (𝑐‘𝑍))) ≠ 0)
642	639, 585, 640, 641	dmmcand 38469	. . . . . . 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))))
643	607, 632, 597, 611	div23d 10717	. . . . . . . . . . 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)))
644	643	eqcomd 2616	. . . . . . . . . 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))))
645		nfv 1830	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
646		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥)
647	619	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇)
648	11	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
649		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑍 → (𝑐‘𝑡) = (𝑐‘𝑍))
650	185, 649	fveq12d 6109	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)) = ((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍)))
651	650	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑍 → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) = (((𝑆 D^𝑛 (𝐻‘𝑍))‘(𝑐‘𝑍))‘𝑥))
652	645, 646, 598, 647, 648, 606, 651, 632	fprodsplitsn 14559	. . . . . . . . . . . 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)))
653	652	eqcomd 2616	. . . . . . . . . . 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))
654	653	oveq1d 6564	. . . . . . . . . 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))))
655	644, 654	eqtrd 2644	. . . . . . . . 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))))
656	655	oveq2d 6565	. . . . . . . 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)))))
657	598, 375, 377	sylancl 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
658	512	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝜑)
659	354	sselda 3568	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
660	659	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ 𝑇)
661	518, 620	fssd 5970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝑁))
662	661	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝑁))
663	658, 660, 662, 148	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
664	663	adantllr 751	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡)):𝑋⟶ℂ)
665	631	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑥 ∈ 𝑋)
666	664, 665	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
667	657, 666	fprodcl 14521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(((𝑆 D^𝑛 (𝐻‘𝑡))‘(𝑐‘𝑡))‘𝑥) ∈ ℂ)
668	636, 637, 667, 597, 638, 611	divmuldivd 10721	. . . . . . . 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)))))
669	564, 596	mulcomd 9940	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
670		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))
671		nfcv 2751	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(!‘(𝑐‘𝑍))
672	512, 10	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅)
673	649	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑍 → (!‘(𝑐‘𝑡)) = (!‘(𝑐‘𝑍)))
674	670, 671, 586, 619, 672, 595, 673, 564	fprodsplitsn 14559	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) = (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))))
675	674	eqcomd 2616	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡)) · (!‘(𝑐‘𝑍))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
676	669, 675	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((!‘(𝑐‘𝑍)) · ∏𝑡 ∈ 𝑅 (!‘(𝑐‘𝑡))) = ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)))
677	676	oveq2d 6565	. . . . . . . . . 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))))
678	677	adantlr 747	. . . . . . . . 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))))
679	512, 378	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑅 ∪ {𝑍}) ∈ Fin)
680	587	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0...𝐽) ⊆ ℕ₀)
681	518	ffvelrnda 6267	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ (0...𝐽))
682	680, 681	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) ∈ ℕ₀)
683	682	faccld 12933	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℕ)
684	683	nncnd 10913	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ∈ ℂ)
685	679, 684	fprodcl 14521	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
686	685	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ∈ ℂ)
687	683	nnne0d 10942	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (!‘(𝑐‘𝑡)) ≠ 0)
688	679, 684, 687	fprodn0 14548	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
689	688	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∏𝑡 ∈ (𝑅 ∪ {𝑍})(!‘(𝑐‘𝑡)) ≠ 0)
690	636, 667, 686, 689	div23d 10717	. . . . . . . . 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)))
691		eqidd 2611	. . . . . . . . 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)))
692	678, 690, 691	3eqtrd 2648	. . . . . . . 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)))
693	656, 668, 692	3eqtrd 2648	. . . . . . 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)))
694	635, 642, 693	3eqtrd 2648	. . . . . 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)))
695	694	sumeq2dv 14281	. . . . 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)))
696	511, 695	eqtrd 2644	. . . 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)))
697	299, 323, 696	3eqtrd 2648	. . 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)))
698	697	mpteq2dva 4672	. 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))))
699	39, 215, 698	3eqtrd 2648	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 {cpr 4127 ⟨cop 4131 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 · cmul 9820 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 !cfa 12922 Ccbc 12951 Σcsu 14264 ∏cprod 14474 ↾_t crest 15904 TopOpenctopn 15905 ℂ_fldccnfld 19567 D^𝑛 cdvn 23434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-prod 14475 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-dvn 23438

This theorem is referenced by: dvnprodlem3 38838

W3C validator