Theorem fsumrelem 15765

Description: Lemma for fsumre 15766, fsumim 15767, and fsumcj 15768. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
fsumre.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsumre.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
fsumrelem.3	⊢ 𝐹:ℂ⟶ℂ
fsumrelem.4	⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))

Assertion

Ref	Expression
fsumrelem	⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))

Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝑥,𝐵,𝑦   𝑘,𝐹,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsumrelem

Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0cn 11131	. . . . . . . 8 ⊢ 0 ∈ ℂ
2		fsumrelem.3	. . . . . . . . 9 ⊢ 𝐹:ℂ⟶ℂ
3	2	ffvelcdmi 7031	. . . . . . . 8 ⊢ (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
4	1, 3	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) ∈ ℂ
5	4	addridi 11328	. . . . . 6 ⊢ ((𝐹‘0) + 0) = (𝐹‘0)
6		fvoveq1 7385	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7		fveq2 6836	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
8	7	oveq1d 7377	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘𝑦)))
9	6, 8	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹‘𝑦))))
10		oveq2 7370	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11		00id 11316	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
12	10, 11	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (0 + 𝑦) = 0)
13	12	fveq2d 6840	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14		fveq2 6836	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
15	14	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹‘0) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘0)))
16	13, 15	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑦 = 0 → ((𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹‘𝑦)) ↔ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))))
17		fsumrelem.4	. . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
18	9, 16, 17	vtocl2ga 3522	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
19	1, 1, 18	mp2an 693	. . . . . 6 ⊢ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
20	5, 19	eqtr2i 2761	. . . . 5 ⊢ ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
21	4, 4, 1	addcani 11334	. . . . 5 ⊢ (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
22	20, 21	mpbi 230	. . . 4 ⊢ (𝐹‘0) = 0
23		sumeq1 15646	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24		sum0 15678	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
25	23, 24	eqtrdi 2788	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
26	25	fveq2d 6840	. . . 4 ⊢ (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = (𝐹‘0))
27		sumeq1 15646	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = Σ𝑘 ∈ ∅ (𝐹‘𝐵))
28		sum0 15678	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐹‘𝐵) = 0
29	27, 28	eqtrdi 2788	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = 0)
30	22, 26, 29	3eqtr4a 2798	. . 3 ⊢ (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
31	30	a1i 11	. 2 ⊢ (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
32		addcl 11115	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
33	32	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
34		fsumre.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
35	34	fmpttd 7063	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
36	35	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
37		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
38		f1of 6776	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
40		fco 6688	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
41	36, 39, 40	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
42	41	ffvelcdmda 7032	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
44		nnuz 12822	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
45	43, 44	eleqtrdi 2847	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
46	17	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
47	39	ffvelcdmda 7032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑥) ∈ 𝐴)
48		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
49		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	49	fvmpt2 6955	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
51	48, 34, 50	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
52	51	fveq2d 6840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘𝐵))
53		fvex 6849	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐵) ∈ V
54		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))
55	54	fvmpt2 6955	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ V) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
56	48, 53, 55	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
57	52, 56	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
58	57	ralrimiva 3130	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
59	58	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
60		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐹
61		nffvmpt1 6847	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))
62	60, 61	nffv 6846	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
63		nffvmpt1 6847	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
64	62, 63	nfeq 2913	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
65		2fveq3 6841	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))))
66		fveq2 6836	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
67	65, 66	eqeq12d 2753	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑥) → ((𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) ↔ (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))))
68	64, 67	rspc 3553	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))))
69	47, 59, 68	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
70		fvco3 6935	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
71	39, 70	sylan 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
72	71	fveq2d 6840	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))))
73		fvco3 6935	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
74	39, 73	sylan 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
75	69, 72, 74	3eqtr4d 2782	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥)) = (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥))
76	33, 42, 45, 46, 75	seqhomo 14006	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘(seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
77		fveq2 6836	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
78	36	ffvelcdmda 7032	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
79	77, 43, 37, 78, 71	fsum 15677	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)))
80	79	fveq2d 6840	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))))
81		fveq2 6836	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
82	2	ffvelcdmi 7031	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (𝐹‘𝐵) ∈ ℂ)
83	34, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
84	83	fmpttd 7063	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
85	84	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
86	85	ffvelcdmda 7032	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) ∈ ℂ)
87	81, 43, 37, 86, 74	fsum 15677	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
88	76, 80, 87	3eqtr4d 2782	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚))
89		sumfc 15666	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
90	89	fveq2i 6839	. . . . . 6 ⊢ (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐹‘Σ𝑘 ∈ 𝐴 𝐵)
91		sumfc 15666	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)
92	88, 90, 91	3eqtr3g 2795	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
93	92	expr 456	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
94	93	exlimdv 1935	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
95	94	expimpd 453	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
96		fsumre.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
97		fz1f1o 15667	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
98	96, 97	syl 17	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
99	31, 95, 98	mpjaod 861	1 ⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ∅c0 4274   ↦ cmpt 5167   ∘ ccom 5630  ⟶wf 6490  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362  Fincfn 8888  ℂcc 11031  0cc0 11033  1c1 11034   + caddc 11036  ℕcn 12169  ℤ_≥cuz 12783  ...cfz 13456  seqcseq 13958  ♯chash 14287  Σcsu 15643

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-sum 15644

This theorem is referenced by:  fsumre  15766  fsumim  15767  fsumcj  15768