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Theorem dvmptfsum 26033

Description: Function-builder for derivative, finite sums rule. (Contributed by Stefan O'Rear, 12-Nov-2014.)

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

**Assertion**
Ref	Expression
dvmptfsum	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))

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

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

Step	Hyp	Ref	Expression
1		ssid 4031	. 2 ⊢ 𝐼 ⊆ 𝐼
2		dvmptfsum.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
3		sseq1 4034	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
4		sumeq1 15737	. . . . . . . . 9 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ ∅ 𝐴)
5	4	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴))
6	5	oveq2d 7464	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)))
7		sumeq1 15737	. . . . . . . 8 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ ∅ 𝐵)
8	7	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))
9	6, 8	eqeq12d 2756	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)))
10	3, 9	imbi12d 344	. . . . 5 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))))
11	10	imbi2d 340	. . . 4 ⊢ (𝑎 = ∅ → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)))))
12		sseq1 4034	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼))
13		sumeq1 15737	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝑏 𝐴)
14	13	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴))
15	14	oveq2d 7464	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)))
16		sumeq1 15737	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝑏 𝐵)
17	16	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))
18	15, 17	eqeq12d 2756	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)))
19	12, 18	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝑏 → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))))
20	19	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)))))
21		sseq1 4034	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
22		sumeq1 15737	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
23	22	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
24	23	oveq2d 7464	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
25		sumeq1 15737	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)
26	25	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))
27	24, 26	eqeq12d 2756	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))
28	21, 27	imbi12d 344	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))
29	28	imbi2d 340	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))))
30		sseq1 4034	. . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼))
31		sumeq1 15737	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝐼 𝐴)
32	31	mpteq2dv 5268	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴))
33	32	oveq2d 7464	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)))
34		sumeq1 15737	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝐼 𝐵)
35	34	mpteq2dv 5268	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))
36	33, 35	eqeq12d 2756	. . . . . 6 ⊢ (𝑎 = 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))
37	30, 36	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐼 → ((𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵)) ↔ (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))))
38	37	imbi2d 340	. . . 4 ⊢ (𝑎 = 𝐼 → ((𝜑 → (𝑎 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵))) ↔ (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))))
39		dvmptfsum.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
40		0cnd 11283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
41		0cnd 11283	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
42	39, 41	dvmptc 26016	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 0)) = (𝑥 ∈ 𝑆 ↦ 0))
43		dvmptfsum.j	. . . . . . . . 9 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
44		dvmptfsum.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
45	44	cnfldtopon 24824	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
46		recnprss 25959	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
47	39, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
48		resttopon 23190	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
49	45, 47, 48	sylancr 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
50	43, 49	eqeltrid 2848	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
51		dvmptfsum.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
52		toponss 22954	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑆) ∧ 𝑋 ∈ 𝐽) → 𝑋 ⊆ 𝑆)
53	50, 51, 52	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
54	39, 40, 40, 42, 53, 43, 44, 51	dvmptres 26021	. . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
55		sum0 15769	. . . . . . . 8 ⊢ Σ𝑖 ∈ ∅ 𝐴 = 0
56	55	mpteq2i 5271	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 0)
57	56	oveq2i 7459	. . . . . 6 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0))
58		sum0 15769	. . . . . . 7 ⊢ Σ𝑖 ∈ ∅ 𝐵 = 0
59	58	mpteq2i 5271	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0)
60	54, 57, 59	3eqtr4g 2805	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))
61	60	a1d 25	. . . 4 ⊢ (𝜑 → (∅ ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵)))
62		ssun1 4201	. . . . . . . . . 10 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
63		sstr 4017	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼)
64	62, 63	mpan 689	. . . . . . . . 9 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑏 ⊆ 𝐼)
65	64	imim1i 63	. . . . . . . 8 ⊢ ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)))
66		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝜑)
67	66, 39	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑆 ∈ {ℝ, ℂ})
68	2	ad3antrrr 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝐼 ∈ Fin)
69	64	ad2antlr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ⊆ 𝐼)
70	68, 69	ssfid 9329	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ∈ Fin)
71		simp-4l 782	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝜑)
72	69	sselda 4008	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑖 ∈ 𝐼)
73		simplr 768	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑎 ∈ 𝑋)
74		nfv 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)
75		nfcsb1v 3946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴
76	75	nfel1 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
77	74, 76	nfim 1895	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
78		eleq1w 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝑋 ↔ 𝑎 ∈ 𝑋))
79	78	3anbi3d 1442	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)))
80		csbeq1a 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑥⦌𝐴)
81	80	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
82	79, 81	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)))
83		dvmptfsum.a	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
84	77, 82, 83	chvarfv 2241	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
85	71, 72, 73, 84	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
86	70, 85	fsumcl 15781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
87	86	adantlrr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
88		sumex 15736	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 ∈ V
89	88	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 ∈ V)
90		nfcv 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐴
91		nfcv 2908	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑏
92	91, 75	nfsum 15739	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴
93	80	sumeq2sdv 15751	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐴 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
94	90, 92, 93	cbvmpt 5277	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
95	94	oveq2i 7459	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴))
96		nfcv 2908	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐵
97		nfcsb1v 3946	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
98	91, 97	nfsum 15739	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵
99		csbeq1a 3935	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
100	99	sumeq2sdv 15751	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐵 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
101	96, 98, 100	cbvmpt 5277	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
102	95, 101	eqeq12i 2758	. . . . . . . . . . . . . 14 ⊢ ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) ↔ (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵))
103	102	biimpi 216	. . . . . . . . . . . . 13 ⊢ ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) → (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵))
104	103	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵))
105		simplll 774	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝜑)
106		ssun2 4202	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐} ⊆ (𝑏 ∪ {𝑐})
107		sstr 4017	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑐} ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → {𝑐} ⊆ 𝐼)
108	106, 107	mpan 689	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → {𝑐} ⊆ 𝐼)
109		vex 3492	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
110	109	snss 4810	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐼 ↔ {𝑐} ⊆ 𝐼)
111	108, 110	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼)
112	111	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑐 ∈ 𝐼)
113		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋)
114	83	3expb 1120	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ ℂ)
115	114	ancom2s 649	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐴 ∈ ℂ)
116	115	ralrimivva 3208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ)
117		nfcsb1v 3946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
118	117	nfel1 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
119		csbeq1a 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
120	119	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
121	76, 118, 81, 120	rspc2 3644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
122	121	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
123	116, 122	mpan9 506	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
124	105, 112, 113, 123	syl12anc 836	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
125	124	adantlrr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
126		dvmptfsum.b	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
127	126	3expb 1120	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐵 ∈ ℂ)
128	127	ancom2s 649	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐵 ∈ ℂ)
129	128	ralrimivva 3208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ)
130	97	nfel1 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
131		nfcsb1v 3946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
132	131	nfel1 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
133	99	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
134		csbeq1a 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
135	134	eleq1d 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
136	130, 132, 133, 135	rspc2 3644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
137	136	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
138	129, 137	mpan9 506	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
139	105, 112, 113, 138	syl12anc 836	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
140	139	adantlrr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
141	111	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑐 ∈ 𝐼)
142		nfv 1913	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑐 ∈ 𝐼)
143		nfcv 2908	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑆
144		nfcv 2908	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖 D
145		nfcv 2908	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝑋
146		nfcsb1v 3946	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴
147	145, 146	nfmpt 5273	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)
148	143, 144, 147	nfov 7478	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
149		nfcsb1v 3946	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐵
150	145, 149	nfmpt 5273	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
151	148, 150	nfeq 2922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
152	142, 151	nfim 1895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
153		eleq1w 2827	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑖 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼))
154	153	anbi2d 629	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑐 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼)))
155		csbeq1a 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
156	155	mpteq2dv 5268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
157	156	oveq2d 7464	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)))
158		csbeq1a 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → 𝐵 = ⦋𝑐 / 𝑖⦌𝐵)
159	158	mpteq2dv 5268	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
160	157, 159	eqeq12d 2756	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑐 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)))
161	154, 160	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑐 → (((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) ↔ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))))
162		dvmptfsum.d	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
163	152, 161, 162	chvarfv 2241	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
164		nfcv 2908	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐴
165		nfcv 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑐
166	165, 75	nfcsbw 3948	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
167	80	csbeq2dv 3928	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
168	164, 166, 167	cbvmpt 5277	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
169	168	oveq2i 7459	. . . . . . . . . . . . . 14 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))
170		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐵
171	165, 97	nfcsbw 3948	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
172	99	csbeq2dv 3928	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
173	170, 171, 172	cbvmpt 5277	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
174	163, 169, 173	3eqtr3g 2803	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))
175	66, 141, 174	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))
176	67, 87, 89, 104, 125, 140, 175	dvmptadd 26018	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)))
177		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴
178		nfcv 2908	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑏 ∪ {𝑐})
179	178, 75	nfsum 15739	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴
180	80	sumeq2sdv 15751	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
181	177, 179, 180	cbvmpt 5277	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
182		simpllr 775	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ¬ 𝑐 ∈ 𝑏)
183		disjsn 4736	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏)
184	182, 183	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∩ {𝑐}) = ∅)
185		eqidd 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) = (𝑏 ∪ {𝑐}))
186		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
187	68, 186	ssfid 9329	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ∈ Fin)
188		simp-4l 782	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝜑)
189	186	sselda 4008	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑖 ∈ 𝐼)
190		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑎 ∈ 𝑋)
191	188, 189, 190, 84	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
192	184, 185, 187, 191	fsumsplit 15789	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴))
193		sumsns 15798	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
194	109, 124, 193	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
195	194	oveq2d 7464	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴) = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))
196	192, 195	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))
197	196	mpteq2dva 5266	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)))
198	181, 197	eqtrid 2792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)))
199	198	adantrr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)))
200	199	oveq2d 7464	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))))
201		nfcv 2908	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵
202	178, 97	nfsum 15739	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵
203	99	sumeq2sdv 15751	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
204	201, 202, 203	cbvmpt 5277	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
205	74, 130	nfim 1895	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
206	79, 133	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)))
207	205, 206, 126	chvarfv 2241	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
208	188, 189, 190, 207	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
209	184, 185, 187, 208	fsumsplit 15789	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵))
210		sumsns 15798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
211	109, 139, 210	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
212	211	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵) = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))
213	209, 212	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵 = (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))
214	213	mpteq2dva 5266	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)))
215	204, 214	eqtrid 2792	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)))
216	215	adantrr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ (Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 + ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)))
217	176, 200, 216	3eqtr4d 2790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))
218	217	exp32 420	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))
219	218	a2d 29	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → (((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))
220	65, 219	syl5 34	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))))
221	220	expcom 413	. . . . . 6 ⊢ (¬ 𝑐 ∈ 𝑏 → (𝜑 → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))))
222	221	adantl 481	. . . . 5 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → (𝜑 → ((𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵)) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))))
223	222	a2d 29	. . . 4 ⊢ ((𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏) → ((𝜑 → (𝑏 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → (𝜑 → ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)))))
224	11, 20, 29, 38, 61, 223	findcard2s 9231	. . 3 ⊢ (𝐼 ∈ Fin → (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))))
225	2, 224	mpcom 38	. 2 ⊢ (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))
226	1, 225	mpi 20	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⦋csb 3921 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 {cpr 4650 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 Σcsu 15734 ↾_t crest 17480 TopOpenctopn 17481 ℂ_fldccnfld 21387 TopOnctopon 22937 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: dvply1 26343 dvtaylp 26430 pserdvlem2 26490 advlogexp 26715 dvnmul 45864 dirkeritg 46023 etransclem2 46157

W3C validator