Theorem fsumrelem 15223

Description: Lemma for fsumre 15224, fsumim 15225, and fsumcj 15226. (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 10684	. . . . . . . 8 ⊢ 0 ∈ ℂ
2		fsumrelem.3	. . . . . . . . 9 ⊢ 𝐹:ℂ⟶ℂ
3	2	ffvelrni 6847	. . . . . . . 8 ⊢ (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
4	1, 3	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) ∈ ℂ
5	4	addid1i 10878	. . . . . 6 ⊢ ((𝐹‘0) + 0) = (𝐹‘0)
6		fvoveq1 7179	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
8	7	oveq1d 7171	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘𝑦)))
9	6, 8	eqeq12d 2774	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹‘𝑦))))
10		oveq2 7164	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11		00id 10866	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
12	10, 11	eqtrdi 2809	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (0 + 𝑦) = 0)
13	12	fveq2d 6667	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
15	14	oveq2d 7172	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹‘0) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘0)))
16	13, 15	eqeq12d 2774	. . . . . . . 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 3495	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
19	1, 1, 18	mp2an 691	. . . . . 6 ⊢ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
20	5, 19	eqtr2i 2782	. . . . 5 ⊢ ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
21	4, 4, 1	addcani 10884	. . . . 5 ⊢ (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
22	20, 21	mpbi 233	. . . 4 ⊢ (𝐹‘0) = 0
23		sumeq1 15106	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24		sum0 15139	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
25	23, 24	eqtrdi 2809	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
26	25	fveq2d 6667	. . . 4 ⊢ (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = (𝐹‘0))
27		sumeq1 15106	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = Σ𝑘 ∈ ∅ (𝐹‘𝐵))
28		sum0 15139	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐹‘𝐵) = 0
29	27, 28	eqtrdi 2809	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = 0)
30	22, 26, 29	3eqtr4a 2819	. . 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 10670	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
33	32	adantl 485	. . . . . . . 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 6876	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
36	35	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
37		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
38		f1of 6607	. . . . . . . . . . 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 6521	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
41	36, 39, 40	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
42	41	ffvelrnda 6848	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
44		nnuz 12334	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
45	43, 44	eleqtrdi 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
46	17	adantl 485	. . . . . . . 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 6848	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑥) ∈ 𝐴)
48		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
49		eqid 2758	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	49	fvmpt2 6775	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
51	48, 34, 50	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
52	51	fveq2d 6667	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘𝐵))
53		fvex 6676	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐵) ∈ V
54		eqid 2758	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))
55	54	fvmpt2 6775	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ V) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
56	48, 53, 55	sylancl 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
57	52, 56	eqtr4d 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
58	57	ralrimiva 3113	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
59	58	ad2antrr 725	. . . . . . . . . 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 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐹
61		nffvmpt1 6674	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))
62	60, 61	nffv 6673	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
63		nffvmpt1 6674	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
64	62, 63	nfeq 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
65		2fveq3 6668	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))))
66		fveq2 6663	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
67	65, 66	eqeq12d 2774	. . . . . . . . . . 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 3531	. . . . . . . . . 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 6756	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
71	39, 70	sylan 583	. . . . . . . . . 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 6667	. . . . . . . . 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 6756	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
74	39, 73	sylan 583	. . . . . . . . 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 2803	. . . . . . . 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 13480	. . . . . . 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 6663	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
78	36	ffvelrnda 6848	. . . . . . . . 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 15138	. . . . . . . 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 6667	. . . . . . 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 6663	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
82	2	ffvelrni 6847	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (𝐹‘𝐵) ∈ ℂ)
83	34, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
84	83	fmpttd 6876	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
85	84	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
86	85	ffvelrnda 6848	. . . . . . . 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 15138	. . . . . . 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 2803	. . . . . 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 15127	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
90	89	fveq2i 6666	. . . . . 6 ⊢ (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐹‘Σ𝑘 ∈ 𝐴 𝐵)
91		sumfc 15127	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)
92	88, 90, 91	3eqtr3g 2816	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
93	92	expr 460	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
94	93	exlimdv 1934	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
95	94	expimpd 457	. 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 15128	. . 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 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  Vcvv 3409  ∅c0 4227   ↦ cmpt 5116   ∘ ccom 5532  ⟶wf 6336  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156  Fincfn 8540  ℂcc 10586  0cc0 10588  1c1 10589   + caddc 10591  ℕcn 11687  ℤ_≥cuz 12295  ...cfz 12952  seqcseq 13431  ♯chash 13753  Σcsu 15103

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-inf2 9150  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-sup 8952  df-oi 9020  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-rp 12444  df-fz 12953  df-fzo 13096  df-seq 13432  df-exp 13493  df-hash 13754  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656  df-clim 14906  df-sum 15104

This theorem is referenced by:  fsumre  15224  fsumim  15225  fsumcj  15226