dvmptfsum - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvmptfsum		Structured version Visualization version GIF version

Theorem dvmptfsum 25995

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 4001	. 2 ⊢ 𝐼 ⊆ 𝐼
2		dvmptfsum.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
3		sseq1 4004	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
4		sumeq1 15688	. . . . . . . . 9 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ ∅ 𝐴)
5	4	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴))
6	5	oveq2d 7432	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)))
7		sumeq1 15688	. . . . . . . 8 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ ∅ 𝐵)
8	7	mpteq2dv 5247	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))
9	6, 8	eqeq12d 2742	. . . . . 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 343	. . . . 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 339	. . . 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 4004	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼))
13		sumeq1 15688	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝑏 𝐴)
14	13	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴))
15	14	oveq2d 7432	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)))
16		sumeq1 15688	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝑏 𝐵)
17	16	mpteq2dv 5247	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))
18	15, 17	eqeq12d 2742	. . . . . 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 343	. . . . 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 339	. . . 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 4004	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
22		sumeq1 15688	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
23	22	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
24	23	oveq2d 7432	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
25		sumeq1 15688	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)
26	25	mpteq2dv 5247	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))
27	24, 26	eqeq12d 2742	. . . . . 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 343	. . . . 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 339	. . . 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 4004	. . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼))
31		sumeq1 15688	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝐼 𝐴)
32	31	mpteq2dv 5247	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴))
33	32	oveq2d 7432	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)))
34		sumeq1 15688	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝐼 𝐵)
35	34	mpteq2dv 5247	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))
36	33, 35	eqeq12d 2742	. . . . . 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 343	. . . . 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 339	. . . 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 11248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
41		0cnd 11248	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
42	39, 41	dvmptc 25978	. . . . . . 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 24787	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
46		recnprss 25921	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
47	39, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
48		resttopon 23153	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
49	45, 47, 48	sylancr 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
50	43, 49	eqeltrid 2830	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
51		dvmptfsum.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
52		toponss 22917	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑆) ∧ 𝑋 ∈ 𝐽) → 𝑋 ⊆ 𝑆)
53	50, 51, 52	syl2anc 582	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
54	39, 40, 40, 42, 53, 43, 44, 51	dvmptres 25983	. . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
55		sum0 15720	. . . . . . . 8 ⊢ Σ𝑖 ∈ ∅ 𝐴 = 0
56	55	mpteq2i 5250	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 0)
57	56	oveq2i 7427	. . . . . 6 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0))
58		sum0 15720	. . . . . . 7 ⊢ Σ𝑖 ∈ ∅ 𝐵 = 0
59	58	mpteq2i 5250	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0)
60	54, 57, 59	3eqtr4g 2791	. . . . 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 4170	. . . . . . . . . 10 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
63		sstr 3987	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼)
64	62, 63	mpan 688	. . . . . . . . 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 765	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝐼 ∈ Fin)
69	64	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ⊆ 𝐼)
70	68, 69	ssfid 9294	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ∈ Fin)
71		simp-4l 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝜑)
72	69	sselda 3978	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑖 ∈ 𝐼)
73		simplr 767	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑎 ∈ 𝑋)
74		nfv 1910	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)
75		nfcsb1v 3916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴
76	75	nfel1 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
77	74, 76	nfim 1892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
78		eleq1w 2809	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝑋 ↔ 𝑎 ∈ 𝑋))
79	78	3anbi3d 1439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)))
80		csbeq1a 3905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑥⦌𝐴)
81	80	eleq1d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
82	79, 81	imbi12d 343	. . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
85	71, 72, 73, 84	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
86	70, 85	fsumcl 15732	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
87	86	adantlrr 719	. . . . . . . . . . . 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 15687	. . . . . . . . . . . . 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 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐴
91		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑏
92	91, 75	nfsum 15690	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴
93	80	sumeq2sdv 15703	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐴 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
94	90, 92, 93	cbvmpt 5256	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
95	94	oveq2i 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴))
96		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐵
97		nfcsb1v 3916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
98	91, 97	nfsum 15690	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵
99		csbeq1a 3905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
100	99	sumeq2sdv 15703	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐵 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
101	96, 98, 100	cbvmpt 5256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
102	95, 101	eqeq12i 2744	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 773	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝜑)
106		ssun2 4171	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐} ⊆ (𝑏 ∪ {𝑐})
107		sstr 3987	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑐} ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → {𝑐} ⊆ 𝐼)
108	106, 107	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → {𝑐} ⊆ 𝐼)
109		vex 3466	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
110	109	snss 4784	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐼 ↔ {𝑐} ⊆ 𝐼)
111	108, 110	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼)
112	111	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑐 ∈ 𝐼)
113		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋)
114	83	3expb 1117	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ ℂ)
115	114	ancom2s 648	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐴 ∈ ℂ)
116	115	ralrimivva 3191	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ)
117		nfcsb1v 3916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
118	117	nfel1 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
119		csbeq1a 3905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
120	119	eleq1d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
121	76, 118, 81, 120	rspc2 3616	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
122	121	ancoms 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
123	116, 122	mpan9 505	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
124	105, 112, 113, 123	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
125	124	adantlrr 719	. . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐵 ∈ ℂ)
128	127	ancom2s 648	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐵 ∈ ℂ)
129	128	ralrimivva 3191	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ)
130	97	nfel1 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
131		nfcsb1v 3916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
132	131	nfel1 2909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
133	99	eleq1d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
134		csbeq1a 3905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
135	134	eleq1d 2811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
136	130, 132, 133, 135	rspc2 3616	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
137	136	ancoms 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
138	129, 137	mpan9 505	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
139	105, 112, 113, 138	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
140	139	adantlrr 719	. . . . . . . . . . . 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 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑐 ∈ 𝐼)
142		nfv 1910	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑐 ∈ 𝐼)
143		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑆
144		nfcv 2892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖 D
145		nfcv 2892	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝑋
146		nfcsb1v 3916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴
147	145, 146	nfmpt 5252	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)
148	143, 144, 147	nfov 7446	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
149		nfcsb1v 3916	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐵
150	145, 149	nfmpt 5252	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
151	148, 150	nfeq 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
152	142, 151	nfim 1892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
153		eleq1w 2809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑖 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼))
154	153	anbi2d 628	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑐 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼)))
155		csbeq1a 3905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
156	155	mpteq2dv 5247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
157	156	oveq2d 7432	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)))
158		csbeq1a 3905	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → 𝐵 = ⦋𝑐 / 𝑖⦌𝐵)
159	158	mpteq2dv 5247	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
160	157, 159	eqeq12d 2742	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
164		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐴
165		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑐
166	165, 75	nfcsbw 3918	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
167	80	csbeq2dv 3898	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
168	164, 166, 167	cbvmpt 5256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
169	168	oveq2i 7427	. . . . . . . . . . . . . 14 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))
170		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐵
171	165, 97	nfcsbw 3918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
172	99	csbeq2dv 3898	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
173	170, 171, 172	cbvmpt 5256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
174	163, 169, 173	3eqtr3g 2789	. . . . . . . . . . . . 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 582	. . . . . . . . . . . 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 25980	. . . . . . . . . . 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 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴
178		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑏 ∪ {𝑐})
179	178, 75	nfsum 15690	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴
180	80	sumeq2sdv 15703	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
181	177, 179, 180	cbvmpt 5256	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
182		simpllr 774	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ¬ 𝑐 ∈ 𝑏)
183		disjsn 4710	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏)
184	182, 183	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∩ {𝑐}) = ∅)
185		eqidd 2727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) = (𝑏 ∪ {𝑐}))
186		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
187	68, 186	ssfid 9294	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ∈ Fin)
188		simp-4l 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝜑)
189	186	sselda 3978	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑖 ∈ 𝐼)
190		simplr 767	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑎 ∈ 𝑋)
191	188, 189, 190, 84	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
192	184, 185, 187, 191	fsumsplit 15740	. . . . . . . . . . . . . . . 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 15749	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
194	109, 124, 193	sylancr 585	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
195	194	oveq2d 7432	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 5245	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 7432	. . . . . . . . . . 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 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵
202	178, 97	nfsum 15690	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵
203	99	sumeq2sdv 15703	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
204	201, 202, 203	cbvmpt 5256	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
205	74, 130	nfim 1892	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
206	79, 133	imbi12d 343	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)))
207	205, 206, 126	chvarfv 2229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
208	188, 189, 190, 207	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
209	184, 185, 187, 208	fsumsplit 15740	. . . . . . . . . . . . . . 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 15749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
211	109, 139, 210	sylancr 585	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
212	211	oveq2d 7432	. . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . 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 5245	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 715	. . . . . . . . . . 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 2776	. . . . . . . . . 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 419	. . . . . . . . 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 412	. . . . . 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 480	. . . . 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 9195	. . 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 Vcvv 3462 ⦋csb 3891 ∪ cun 3944 ∩ cin 3945 ⊆ wss 3946 ∅c0 4322 {csn 4623 {cpr 4625 ↦ cmpt 5228 ‘cfv 6546 (class class class)co 7416 Fincfn 8966 ℂcc 11147 ℝcr 11148 0cc0 11149 + caddc 11152 Σcsu 15685 ↾_t crest 17430 TopOpenctopn 17431 ℂ_fldccnfld 21339 TopOnctopon 22900 D cdv 25880

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9399 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-icc 13379 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-hom 17285 df-cco 17286 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-pt 17454 df-prds 17457 df-xrs 17512 df-qtop 17517 df-imas 17518 df-xps 17520 df-mre 17594 df-mrc 17595 df-acs 17597 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-mulg 19058 df-cntz 19307 df-cmn 19776 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-fbas 21336 df-fg 21337 df-cnfld 21340 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22937 df-cld 23011 df-ntr 23012 df-cls 23013 df-nei 23090 df-lp 23128 df-perf 23129 df-cn 23219 df-cnp 23220 df-haus 23307 df-tx 23554 df-hmeo 23747 df-fil 23838 df-fm 23930 df-flim 23931 df-flf 23932 df-xms 24314 df-ms 24315 df-tms 24316 df-cncf 24886 df-limc 25883 df-dv 25884

This theorem is referenced by: dvply1 26308 dvtaylp 26395 pserdvlem2 26455 advlogexp 26679 dvnmul 45600 dirkeritg 45759 etransclem2 45893

W3C validator