Theorem fsumrelem 15519

Description: Lemma for fsumre 15520, fsumim 15521, and fsumcj 15522. (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 10967	. . . . . . . 8 ⊢ 0 ∈ ℂ
2		fsumrelem.3	. . . . . . . . 9 ⊢ 𝐹:ℂ⟶ℂ
3	2	ffvelrni 6960	. . . . . . . 8 ⊢ (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
4	1, 3	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) ∈ ℂ
5	4	addid1i 11162	. . . . . 6 ⊢ ((𝐹‘0) + 0) = (𝐹‘0)
6		fvoveq1 7298	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
8	7	oveq1d 7290	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘𝑦)))
9	6, 8	eqeq12d 2754	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹‘𝑦))))
10		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11		00id 11150	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
12	10, 11	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (0 + 𝑦) = 0)
13	12	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
15	14	oveq2d 7291	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹‘0) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘0)))
16	13, 15	eqeq12d 2754	. . . . . . . 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 3514	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
19	1, 1, 18	mp2an 689	. . . . . 6 ⊢ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
20	5, 19	eqtr2i 2767	. . . . 5 ⊢ ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
21	4, 4, 1	addcani 11168	. . . . 5 ⊢ (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
22	20, 21	mpbi 229	. . . 4 ⊢ (𝐹‘0) = 0
23		sumeq1 15400	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24		sum0 15433	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
25	23, 24	eqtrdi 2794	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
26	25	fveq2d 6778	. . . 4 ⊢ (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = (𝐹‘0))
27		sumeq1 15400	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = Σ𝑘 ∈ ∅ (𝐹‘𝐵))
28		sum0 15433	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐹‘𝐵) = 0
29	27, 28	eqtrdi 2794	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = 0)
30	22, 26, 29	3eqtr4a 2804	. . 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 10953	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
33	32	adantl 482	. . . . . . . 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 6989	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
36	35	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
37		simprr 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
38		f1of 6716	. . . . . . . . . . 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 6624	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
41	36, 39, 40	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
42	41	ffvelrnda 6961	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
44		nnuz 12621	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
45	43, 44	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
46	17	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
47	39	ffvelrnda 6961	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑥) ∈ 𝐴)
48		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
49		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	49	fvmpt2 6886	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
51	48, 34, 50	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
52	51	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘𝐵))
53		fvex 6787	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐵) ∈ V
54		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))
55	54	fvmpt2 6886	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ V) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
56	48, 53, 55	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
57	52, 56	eqtr4d 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
58	57	ralrimiva 3103	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
59	58	ad2antrr 723	. . . . . . . . . 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 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐹
61		nffvmpt1 6785	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))
62	60, 61	nffv 6784	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
63		nffvmpt1 6785	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
64	62, 63	nfeq 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
65		2fveq3 6779	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))))
66		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
67	65, 66	eqeq12d 2754	. . . . . . . . . . 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 3549	. . . . . . . . . 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 6867	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
71	39, 70	sylan 580	. . . . . . . . . 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 6778	. . . . . . . . 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 6867	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
74	39, 73	sylan 580	. . . . . . . . 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 2788	. . . . . . . 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 13770	. . . . . . 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 6774	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
78	36	ffvelrnda 6961	. . . . . . . . 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 15432	. . . . . . . 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 6778	. . . . . . 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 6774	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
82	2	ffvelrni 6960	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (𝐹‘𝐵) ∈ ℂ)
83	34, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
84	83	fmpttd 6989	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
85	84	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
86	85	ffvelrnda 6961	. . . . . . . 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 15432	. . . . . . 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 2788	. . . . . 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 15421	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
90	89	fveq2i 6777	. . . . . 6 ⊢ (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐹‘Σ𝑘 ∈ 𝐴 𝐵)
91		sumfc 15421	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)
92	88, 90, 91	3eqtr3g 2801	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
93	92	expr 457	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
94	93	exlimdv 1936	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
95	94	expimpd 454	. 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 15422	. . 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 857	1 ⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ∅c0 4256   ↦ cmpt 5157   ∘ ccom 5593  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874  ℕcn 11973  ℤ_≥cuz 12582  ...cfz 13239  seqcseq 13721  ♯chash 14044  Σcsu 15397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398

This theorem is referenced by:  fsumre  15520  fsumim  15521  fsumcj  15522