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Theorem dvmptfsum 23738

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 3624	. 2 ⊢ 𝐼 ⊆ 𝐼
2		dvmptfsum.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ Fin)
3		sseq1 3626	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼))
4		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ ∅ 𝐴)
5	4	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴))
6	5	oveq2d 6666	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)))
7		sumeq1 14419	. . . . . . . 8 ⊢ (𝑎 = ∅ → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ ∅ 𝐵)
8	7	mpteq2dv 4745	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵))
9	6, 8	eqeq12d 2637	. . . . . 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 334	. . . . 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 330	. . . 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 3626	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼))
13		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝑏 𝐴)
14	13	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴))
15	14	oveq2d 6666	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)))
16		sumeq1 14419	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝑏 𝐵)
17	16	mpteq2dv 4745	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))
18	15, 17	eqeq12d 2637	. . . . . 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 334	. . . . 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 330	. . . 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 3626	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ⊆ 𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
22		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
23	22	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
24	23	oveq2d 6666	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
25		sumeq1 14419	. . . . . . . 8 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵)
26	25	mpteq2dv 4745	. . . . . . 7 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵))
27	24, 26	eqeq12d 2637	. . . . . 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 334	. . . . 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 330	. . . 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 3626	. . . . . 6 ⊢ (𝑎 = 𝐼 → (𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼))
31		sumeq1 14419	. . . . . . . . 9 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐴 = Σ𝑖 ∈ 𝐼 𝐴)
32	31	mpteq2dv 4745	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴))
33	32	oveq2d 6666	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)))
34		sumeq1 14419	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → Σ𝑖 ∈ 𝑎 𝐵 = Σ𝑖 ∈ 𝐼 𝐵)
35	34	mpteq2dv 4745	. . . . . . 7 ⊢ (𝑎 = 𝐼 → (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑎 𝐵) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))
36	33, 35	eqeq12d 2637	. . . . . 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 334	. . . . 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 330	. . . 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 10033	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 0 ∈ ℂ)
41		0cnd 10033	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
42	39, 41	dvmptc 23721	. . . . . . 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 22586	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
46		recnprss 23668	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
47	39, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
48		resttopon 20965	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
49	45, 47, 48	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
50	43, 49	syl5eqel 2705	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
51		dvmptfsum.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
52		toponss 20731	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑆) ∧ 𝑋 ∈ 𝐽) → 𝑋 ⊆ 𝑆)
53	50, 51, 52	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
54	39, 40, 40, 42, 53, 43, 44, 51	dvmptres 23726	. . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
55		sum0 14452	. . . . . . . 8 ⊢ Σ𝑖 ∈ ∅ 𝐴 = 0
56	55	mpteq2i 4741	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴) = (𝑥 ∈ 𝑋 ↦ 0)
57	56	oveq2i 6661	. . . . . 6 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0))
58		sum0 14452	. . . . . . 7 ⊢ Σ𝑖 ∈ ∅ 𝐵 = 0
59	58	mpteq2i 4741	. . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ ∅ 𝐵) = (𝑥 ∈ 𝑋 ↦ 0)
60	54, 57, 59	3eqtr4g 2681	. . . . 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 3776	. . . . . . . . . 10 ⊢ 𝑏 ⊆ (𝑏 ∪ {𝑐})
63		sstr 3611	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏 ⊆ 𝐼)
64	62, 63	mpan 706	. . . . . . . . 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 790	. . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝐼 ∈ Fin)
69	64	ad2antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ⊆ 𝐼)
70		ssfi 8180	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ Fin ∧ 𝑏 ⊆ 𝐼) → 𝑏 ∈ Fin)
71	68, 69, 70	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑏 ∈ Fin)
72		simp-4l 806	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝜑)
73	69	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑖 ∈ 𝐼)
74		simplr 792	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → 𝑎 ∈ 𝑋)
75		nfv 1843	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)
76		nfcsb1v 3549	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴
77	76	nfel1 2779	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
78	75, 77	nfim 1825	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
79		eleq1 2689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝑋 ↔ 𝑎 ∈ 𝑋))
80	79	3anbi3d 1405	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) ↔ (𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)))
81		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → 𝐴 = ⦋𝑎 / 𝑥⦌𝐴)
82	81	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐴 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
83	80, 82	imbi12d 334	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)))
84		dvmptfsum.a	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
85	78, 83, 84	chvar 2262	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
86	72, 73, 74, 85	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ 𝑏) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
87	71, 86	fsumcl 14464	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
88	87	adantlrr 757	. . . . . . . . . . . 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)
89		sumex 14418	. . . . . . . . . . . . 13 ⊢ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵 ∈ V
90	89	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)
91		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐴
92		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝑏
93	92, 76	nfsum 14421	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴
94	81	sumeq2sdv 14435	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐴 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
95	91, 93, 94	cbvmpt 4749	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴)
96	95	oveq2i 6661	. . . . . . . . . . . . . . 15 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐴))
97		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎Σ𝑖 ∈ 𝑏 𝐵
98		nfcsb1v 3549	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
99	92, 98	nfsum 14421	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵
100		csbeq1a 3542	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
101	100	sumeq2sdv 14435	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ 𝑏 𝐵 = Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
102	97, 99, 101	cbvmpt 4749	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 ⦋𝑎 / 𝑥⦌𝐵)
103	96, 102	eqeq12i 2636	. . . . . . . . . . . . . 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))
104	103	biimpi 206	. . . . . . . . . . . . 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))
105	104	ad2antll 765	. . . . . . . . . . . 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))
106		simplll 798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝜑)
107		ssun2 3777	. . . . . . . . . . . . . . . . 17 ⊢ {𝑐} ⊆ (𝑏 ∪ {𝑐})
108		sstr 3611	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑐} ⊆ (𝑏 ∪ {𝑐}) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → {𝑐} ⊆ 𝐼)
109	107, 108	mpan 706	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → {𝑐} ⊆ 𝐼)
110		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
111	110	snss 4316	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐼 ↔ {𝑐} ⊆ 𝐼)
112	109, 111	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → 𝑐 ∈ 𝐼)
113	112	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑐 ∈ 𝐼)
114		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ 𝑋)
115	84	3expb 1266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐴 ∈ ℂ)
116	115	ancom2s 844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐴 ∈ ℂ)
117	116	ralrimivva 2971	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ)
118		nfcsb1v 3549	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
119	118	nfel1 2779	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ
120		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
121	120	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
122	77, 119, 82, 121	rspc2 3320	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
123	122	ancoms 469	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐴 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ))
124	117, 123	mpan9 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
125	106, 113, 114, 124	syl12anc 1324	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
126	125	adantlrr 757	. . . . . . . . . . . 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)
127		dvmptfsum.b	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
128	127	3expb 1266	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝐵 ∈ ℂ)
129	128	ancom2s 844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑖 ∈ 𝐼)) → 𝐵 ∈ ℂ)
130	129	ralrimivva 2971	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ)
131	98	nfel1 2779	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
132		nfcsb1v 3549	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
133	132	nfel1 2779	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ
134	100	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ ℂ ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
135		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → ⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
136	135	eleq1d 2686	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ ↔ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
137	131, 133, 134, 136	rspc2 3320	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑐 ∈ 𝐼) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
138	137	ancoms 469	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑖 ∈ 𝐼 𝐵 ∈ ℂ → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ))
139	130, 138	mpan9 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋)) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
140	106, 113, 114, 139	syl12anc 1324	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
141	140	adantlrr 757	. . . . . . . . . . . 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)
142	112	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ ((𝑏 ∪ {𝑐}) ⊆ 𝐼 ∧ (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝑏 𝐵))) → 𝑐 ∈ 𝐼)
143		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑐 ∈ 𝐼)
144		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑆
145		nfcv 2764	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖 D
146		nfcv 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖𝑋
147		nfcsb1v 3549	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐴
148	146, 147	nfmpt 4746	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)
149	144, 145, 148	nfov 6676	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
150		nfcsb1v 3549	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖⦋𝑐 / 𝑖⦌𝐵
151	146, 150	nfmpt 4746	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖(𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
152	149, 151	nfeq 2776	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)
153	143, 152	nfim 1825	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
154		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑖 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼))
155	154	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑐 → ((𝜑 ∧ 𝑖 ∈ 𝐼) ↔ (𝜑 ∧ 𝑐 ∈ 𝐼)))
156		csbeq1a 3542	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑐 → 𝐴 = ⦋𝑐 / 𝑖⦌𝐴)
157	156	mpteq2dv 4745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴))
158	157	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)))
159		csbeq1a 3542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑐 → 𝐵 = ⦋𝑐 / 𝑖⦌𝐵)
160	159	mpteq2dv 4745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑐 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
161	158, 160	eqeq12d 2637	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑐 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵) ↔ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵)))
162	155, 161	imbi12d 334	. . . . . . . . . . . . . . 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))))
163		dvmptfsum.d	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
164	153, 162, 163	chvar 2262	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵))
165		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐴
166		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝑐
167	166, 76	nfcsb 3551	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴
168	81	csbeq2dv 3992	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
169	165, 167, 168	cbvmpt 4749	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
170	169	oveq2i 6661	. . . . . . . . . . . . . 14 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐴)) = (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴))
171		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎⦋𝑐 / 𝑖⦌𝐵
172	166, 98	nfcsb 3551	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵
173	100	csbeq2dv 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → ⦋𝑐 / 𝑖⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
174	171, 172, 173	cbvmpt 4749	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌𝐵) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
175	164, 170, 174	3eqtr3g 2679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐼) → (𝑆 D (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)) = (𝑎 ∈ 𝑋 ↦ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵))
176	66, 142, 175	syl2anc 693	. . . . . . . . . . . 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))
177	67, 88, 90, 105, 126, 141, 176	dvmptadd 23723	. . . . . . . . . . 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)))
178		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴
179		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑏 ∪ {𝑐})
180	179, 76	nfsum 14421	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴
181	81	sumeq2sdv 14435	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
182	178, 180, 181	cbvmpt 4749	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐴) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐴)
183		simpllr 799	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → ¬ 𝑐 ∈ 𝑏)
184		disjsn 4246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∩ {𝑐}) = ∅ ↔ ¬ 𝑐 ∈ 𝑏)
185	183, 184	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∩ {𝑐}) = ∅)
186		eqidd 2623	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) = (𝑏 ∪ {𝑐}))
187		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
188		ssfi 8180	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ Fin ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑏 ∪ {𝑐}) ∈ Fin)
189	68, 187, 188	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → (𝑏 ∪ {𝑐}) ∈ Fin)
190		simp-4l 806	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝜑)
191	187	sselda 3603	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑖 ∈ 𝐼)
192		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → 𝑎 ∈ 𝑋)
193	190, 191, 192, 85	syl3anc 1326	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ)
194	185, 186, 189, 193	fsumsplit 14471	. . . . . . . . . . . . . . . 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))
195		sumsns 14479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
196	110, 125, 195	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐴 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐴)
197	196	oveq2d 6666	. . . . . . . . . . . . . . . 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))
198	194, 197	eqtrd 2656	. . . . . . . . . . . . . . 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))
199	198	mpteq2dva 4744	. . . . . . . . . . . . . 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)))
200	182, 199	syl5eq 2668	. . . . . . . . . . . . 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)))
201	200	adantrr 753	. . . . . . . . . . . 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)))
202	201	oveq2d 6666	. . . . . . . . . . 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))))
203		nfcv 2764	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵
204	179, 98	nfsum 14421	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵
205	100	sumeq2sdv 14435	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵 = Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
206	203, 204, 205	cbvmpt 4749	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})𝐵) = (𝑎 ∈ 𝑋 ↦ Σ𝑖 ∈ (𝑏 ∪ {𝑐})⦋𝑎 / 𝑥⦌𝐵)
207	75, 131	nfim 1825	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
208	80, 134	imbi12d 334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑎 → (((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)))
209	207, 208, 127	chvar 2262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑎 ∈ 𝑋) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
210	190, 191, 192, 209	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) ∧ 𝑖 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ)
211	185, 186, 189, 210	fsumsplit 14471	. . . . . . . . . . . . . . 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))
212		sumsns 14479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ V ∧ ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵 ∈ ℂ) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
213	110, 140, 212	sylancr 695	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑏) ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑎 ∈ 𝑋) → Σ𝑖 ∈ {𝑐}⦋𝑎 / 𝑥⦌𝐵 = ⦋𝑐 / 𝑖⦌⦋𝑎 / 𝑥⦌𝐵)
214	213	oveq2d 6666	. . . . . . . . . . . . . . 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))
215	211, 214	eqtrd 2656	. . . . . . . . . . . . . 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))
216	215	mpteq2dva 4744	. . . . . . . . . . . . 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)))
217	206, 216	syl5eq 2668	. . . . . . . . . . . 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)))
218	217	adantrr 753	. . . . . . . . . . 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)))
219	177, 202, 218	3eqtr4d 2666	. . . . . . . . . 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))
220	219	exp32 631	. . . . . . . . 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))))
221	220	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))))
222	65, 221	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))))
223	222	expcom 451	. . . . . 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)))))
224	223	adantl 482	. . . . 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)))))
225	224	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)))))
226	11, 20, 29, 38, 61, 225	findcard2s 8201	. . 3 ⊢ (𝐼 ∈ Fin → (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵))))
227	2, 226	mpcom 38	. 2 ⊢ (𝜑 → (𝐼 ⊆ 𝐼 → (𝑆 D (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐴)) = (𝑥 ∈ 𝑋 ↦ Σ𝑖 ∈ 𝐼 𝐵)))
228	1, 227	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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⦋csb 3533 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 {csn 4177 {cpr 4179 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 Σcsu 14416 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 TopOnctopon 20715 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631

This theorem is referenced by: dvply1 24039 dvtaylp 24124 pserdvlem2 24182 advlogexp 24401 dvnmul 40158 dirkeritg 40319 etransclem2 40453

W3C validator