dvmptfprod - Mathbox for Glauco Siliprandi

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Theorem dvmptfprod 44176

Description: Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvmptfprod.iph	⊢ Ⅎ𝑖𝜑
dvmptfprod.jph	⊢ Ⅎ𝑗𝜑
dvmptfprod.j	⊢ 𝐽 = (𝐾 ↾_t 𝑆)
dvmptfprod.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvmptfprod.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptfprod.x	⊢ (𝜑 → 𝑋 ∈ 𝐽)
dvmptfprod.i	⊢ (𝜑 → 𝐼 ∈ Fin)
dvmptfprod.a	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptfprod.b	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
dvmptfprod.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
dvmptfprod.bc	⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)

**Assertion**
Ref	Expression
dvmptfprod	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))

Distinct variable groups: 𝐴,𝑗 𝐶,𝑖 𝑖,𝐼,𝑗,𝑥 𝑆,𝑖,𝑗,𝑥 𝑖,𝑋,𝑗,𝑥 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑖,𝑗) 𝐴(𝑥,𝑖) 𝐵(𝑥,𝑖,𝑗) 𝐶(𝑥,𝑗) 𝐽(𝑥,𝑖,𝑗) 𝐾(𝑥,𝑖,𝑗)

Proof of Theorem dvmptfprod Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvmptfprod.i	. 2 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		ssid 3966	. . 3 ⊢ 𝐼 ⊆ 𝐼
3	2	jctr 525	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ⊆ 𝐼))
4		sseq1 3969	. . . . 5 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
5	4	anbi2d 629	. . . 4 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6		prodeq1 15792	. . . . . . 7 ⊢ (𝑎 = ∅ → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
7	6	mpteq2dv 5207	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
8	7	oveq2d 7373	. . . . 5 ⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9		sumeq1 15573	. . . . . . 7 ⊢ (𝑎 = ∅ → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
11	10	prodeq1d 15804	. . . . . . . . 9 ⊢ (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
12	11	oveq2d 7373	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
13	12	sumeq2sdv 15589	. . . . . . 7 ⊢ (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
14	9, 13	eqtrd 2776	. . . . . 6 ⊢ (𝑎 = ∅ → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
15	14	mpteq2dv 5207	. . . . 5 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
16	8, 15	eqeq12d 2752	. . . 4 ⊢ (𝑎 = ∅ → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))))
17	5, 16	imbi12d 344	. . 3 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))))
18		sseq1 3969	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼))
19	18	anbi2d 629	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ 𝑏 ⊆ 𝐼)))
20		prodeq1 15792	. . . . . . 7 ⊢ (𝑎 = 𝑏 → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ 𝑏 𝐴)
21	20	mpteq2dv 5207	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴))
22	21	oveq2d 7373	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)))
23		sumeq1 15573	. . . . . . 7 ⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
25	24	prodeq1d 15804	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
26	25	oveq2d 7373	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
27	26	sumeq2sdv 15589	. . . . . . 7 ⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
28	23, 27	eqtrd 2776	. . . . . 6 ⊢ (𝑎 = 𝑏 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
29	28	mpteq2dv 5207	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
30	22, 29	eqeq12d 2752	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))))
31	19, 30	imbi12d 344	. . 3 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))))
32		sseq1 3969	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
33	32	anbi2d 629	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34		prodeq1 15792	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
35	34	mpteq2dv 5207	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
36	35	oveq2d 7373	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37		sumeq1 15573	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38		difeq1 4075	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
39	38	prodeq1d 15804	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
40	39	oveq2d 7373	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
41	40	sumeq2sdv 15589	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
42	37, 41	eqtrd 2776	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
43	42	mpteq2dv 5207	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
44	36, 43	eqeq12d 2752	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))))
45	33, 44	imbi12d 344	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))))
46		sseq1 3969	. . . . 5 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼))
47	46	anbi2d 629	. . . 4 ⊢ (𝑎 = 𝐼 → ((𝜑 ∧ 𝑎 ⊆ 𝐼) ↔ (𝜑 ∧ 𝐼 ⊆ 𝐼)))
48		prodeq1 15792	. . . . . . 7 ⊢ (𝑎 = 𝐼 → ∏𝑖 ∈ 𝑎 𝐴 = ∏𝑖 ∈ 𝐼 𝐴)
49	48	mpteq2dv 5207	. . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴))
50	49	oveq2d 7373	. . . . 5 ⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)))
51		sumeq1 15573	. . . . . . 7 ⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52		difeq1 4075	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
53	52	prodeq1d 15804	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
54	53	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑎 = 𝐼 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
55	54	a1d 25	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → (𝑗 ∈ 𝐼 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
56	55	ralrimiv 3142	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → ∀𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
57	56	sumeq2d 15587	. . . . . . 7 ⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
58	51, 57	eqtrd 2776	. . . . . 6 ⊢ (𝑎 = 𝐼 → Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
59	58	mpteq2dv 5207	. . . . 5 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
60	50, 59	eqeq12d 2752	. . . 4 ⊢ (𝑎 = 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))))
61	47, 60	imbi12d 344	. . 3 ⊢ (𝑎 = 𝐼 → (((𝜑 ∧ 𝑎 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ 𝐼 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))))
62		prod0 15826	. . . . . . . 8 ⊢ ∏𝑖 ∈ ∅ 𝐴 = 1
63	62	mpteq2i 5210	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 1)
64	63	oveq2i 7368	. . . . . 6 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 1))
65	64	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 1)))
66		dvmptfprod.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
67		dvmptfprod.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
68		dvmptfprod.j	. . . . . . . 8 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
69		dvmptfprod.k	. . . . . . . . 9 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
70	69	oveq1i 7367	. . . . . . . 8 ⊢ (𝐾 ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
71	68, 70	eqtri 2764	. . . . . . 7 ⊢ 𝐽 = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
72	67, 71	eleqtrdi 2848	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
73		1cnd 11150	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
74	66, 72, 73	dvmptconst 44146	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 1)) = (𝑥 ∈ 𝑋 ↦ 0))
75		sum0 15606	. . . . . . . 8 ⊢ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
76	75	eqcomi 2745	. . . . . . 7 ⊢ 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
77	76	mpteq2i 5210	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
78	77	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 0) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
79	65, 74, 78	3eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
80	79	adantr 481	. . 3 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
81		simp3 1138	. . . . 5 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
82		simp1r 1198	. . . . 5 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐 ∈ 𝑏)
83		simpl 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84		ssun1 4132	. . . . . . . . . . 11 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
85		sstr2 3951	. . . . . . . . . . 11 ⊢ (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼))
86	84, 85	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼)
87	86	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼)
88	83, 87	jca 512	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑 ∧ 𝑏 ⊆ 𝐼))
89	88	adantl 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑 ∧ 𝑏 ⊆ 𝐼))
90		simpl 483	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))))
91	89, 90	mpd 15	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
92	91	3adant1 1130	. . . . 5 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
93		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏)
94		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑆
95		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥 D
96		nfmpt1 5213	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)
97	94, 95, 96	nfov 7387	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴))
98		nfmpt1 5213	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
99	97, 98	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑥(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
100	93, 99	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑥(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
101		dvmptfprod.iph	. . . . . . . . 9 ⊢ Ⅎ𝑖𝜑
102		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103	101, 102	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑖 ¬ 𝑐 ∈ 𝑏
105	103, 104	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏)
106		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑖𝑆
107		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑖 D
108		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑋
109		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑏
110	109	nfcprod1 15793	. . . . . . . . . 10 ⊢ Ⅎ𝑖∏𝑖 ∈ 𝑏 𝐴
111	108, 110	nfmpt 5212	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)
112	106, 107, 111	nfov 7387	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴))
113		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝐶
114		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 ·
115		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑏 ∖ {𝑗})
116	115	nfcprod1 15793	. . . . . . . . . . 11 ⊢ Ⅎ𝑖∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117	113, 114, 116	nfov 7387	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118	109, 117	nfsum 15575	. . . . . . . . 9 ⊢ Ⅎ𝑖Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119	108, 118	nfmpt 5212	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120	112, 119	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121	105, 120	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑖(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
122		dvmptfprod.jph	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
123		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124	122, 123	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑗 ¬ 𝑐 ∈ 𝑏
126	124, 125	nfan 1902	. . . . . . 7 ⊢ Ⅎ𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏)
127		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴))
128		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑋
129		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑏
130	129	nfsum1 15574	. . . . . . . . 9 ⊢ Ⅎ𝑗Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131	128, 130	nfmpt 5212	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132	127, 131	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133	126, 132	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134		nfcsb1v 3880	. . . . . 6 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴
135		nfcsb1v 3880	. . . . . 6 ⊢ Ⅎ𝑗⦋𝑐 / 𝑗⦌𝐶
136	83	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
137	136	3ad2ant1 1133	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝜑)
138		simp2 1137	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝑖 ∈ 𝐼)
139		simp3 1138	. . . . . . 7 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
140		dvmptfprod.a	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
141	137, 138, 139, 140	syl3anc 1371	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
142	136, 1	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝐼 ∈ Fin)
143	87	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ⊆ 𝐼)
144		ssfi 9117	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ 𝑏 ⊆ 𝐼) → 𝑏 ∈ Fin)
145	142, 143, 144	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146		vex 3449	. . . . . . 7 ⊢ 𝑐 ∈ V
147	146	a1i 11	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑐 ∈ V)
148		simplr 767	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐 ∈ 𝑏)
149		simpllr 774	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
150	66	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151	136	ad2antrr 724	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝜑)
152	143	ad2antrr 724	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑏 ⊆ 𝐼)
153		simpr 485	. . . . . . . 8 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝑏)
154	152, 153	sseldd 3945	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑗 ∈ 𝐼)
155		simplr 767	. . . . . . 7 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝑥 ∈ 𝑋)
156		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑗 ∈ 𝐼
157		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑥 ∈ 𝑋
158	101, 156, 157	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)
159		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝐶 ∈ ℂ
160	158, 159	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
161		eleq1w 2820	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝐼 ↔ 𝑗 ∈ 𝐼))
162	161	3anbi2d 1441	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)))
163		dvmptfprod.bc	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)
164	163	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165	162, 164	imbi12d 344	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)))
166		dvmptfprod.b	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
167	160, 165, 166	chvarfv 2233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ)
168	151, 154, 155, 167	syl3anc 1371	. . . . . 6 ⊢ ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) ∧ 𝑗 ∈ 𝑏) → 𝐶 ∈ ℂ)
169		simpr 485	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
170	83	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝜑)
171		id 22	. . . . . . . . . . 11 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172	146	snid 4622	. . . . . . . . . . . . 13 ⊢ 𝑐 ∈ {𝑐}
173		elun2 4137	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174	172, 173	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝑐 ∈ (𝑏 ∪ {𝑐})
175	174	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ (𝑏 ∪ {𝑐}))
176	171, 175	sseldd 3945	. . . . . . . . . 10 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼)
177	176	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝑐 ∈ 𝐼)
178		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
179		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑐 ∈ 𝐼
180		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 ∈ 𝑋
181	122, 179, 180	nf3an 1904	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)
182		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗ℂ
183	135, 182	nfel 2921	. . . . . . . . . . 11 ⊢ Ⅎ𝑗⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ
184	181, 183	nfim 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)
185		eleq1w 2820	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑐 → (𝑗 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼))
186	185	3anbi2d 1441	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑐 → ((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)))
187		csbeq1a 3869	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑗⦌𝐶)
188	187	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ))
189	186, 188	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑗 = 𝑐 → (((𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)))
190	184, 189, 167	chvarfv 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)
191	170, 177, 178, 190	syl3anc 1371	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)
192	191	adantlr 713	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)
193	192	adantlr 713	. . . . . 6 ⊢ (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥 ∈ 𝑋) → ⦋𝑐 / 𝑗⦌𝐶 ∈ ℂ)
194	122, 179	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑐 ∈ 𝐼)
195		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
196	128, 135	nfmpt 5212	. . . . . . . . . . 11 ⊢ Ⅎ𝑗(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)
197	195, 196	nfeq 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)
198	194, 197	nfim 1899	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))
199	185	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑗 = 𝑐 → ((𝜑 ∧ 𝑗 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼)))
200		csbeq1a 3869	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑐 → ⦋𝑗 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑗⦌⦋𝑗 / 𝑖⦌𝐴)
201		csbcow 3870	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑐 / 𝑗⦌⦋𝑗 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌𝐴
202	201	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑐 → ⦋𝑐 / 𝑗⦌⦋𝑗 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
203	200, 202	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑐 → ⦋𝑗 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
204	203	mpteq2dv 5207	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑐 → (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
205	204	oveq2d 7373	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)))
206	187	mpteq2dv 5207	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐶) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))
207	205, 206	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑗 = 𝑐 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶)))
208	199, 207	imbi12d 344	. . . . . . . . 9 ⊢ (𝑗 = 𝑐 → (((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))))
209	101, 156	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑗 ∈ 𝐼)
210		nfcsb1v 3880	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖⦋𝑗 / 𝑖⦌𝐴
211	108, 210	nfmpt 5212	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)
212	106, 107, 211	nfov 7387	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴))
213		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ 𝐶)
214	212, 213	nfeq 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)
215	209, 214	nfim 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
216	161	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑗 ∈ 𝐼)))
217		csbeq1a 3869	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → 𝐴 = ⦋𝑗 / 𝑖⦌𝐴)
218	217	mpteq2dv 5207	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴))
219	218	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)))
220	163	idi 1	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐶)
221	220	mpteq2dv 5207	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐶))
222	219, 221	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶)))
223	216, 222	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))))
224		dvmptfprod.d	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
225	215, 223, 224	chvarfv 2233	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑗 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐶))
226	198, 208, 225	chvarfv 2233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))
227	176, 226	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))
228	227	ad2antrr 724	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑗⦌𝐶))
229		csbeq1a 3869	. . . . . 6 ⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
230	100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187	dvmptfprodlem 44175	. . . . 5 ⊢ ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
231	81, 82, 92, 230	syl21anc 836	. . . 4 ⊢ (((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
232	231	3exp 1119	. . 3 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (((𝜑 ∧ 𝑏 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))))
233	17, 31, 45, 61, 80, 232	findcard2s 9109	. 2 ⊢ (𝐼 ∈ Fin → ((𝜑 ∧ 𝐼 ⊆ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))))
234	1, 3, 233	sylc 65	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ ∏𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑗 ∈ 𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Vcvv 3445 ⦋csb 3855 ∖ cdif 3907 ∪ cun 3908 ⊆ wss 3910 ∅c0 4282 {csn 4586 {cpr 4588 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7357 Fincfn 8883 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 Σcsu 15570 ∏cprod 15788 ↾_t crest 17302 TopOpenctopn 17303 ℂ_fldccnfld 20796 D cdv 25227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-prod 15789 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231

This theorem is referenced by: (None)

W3C validator