Theorem fsumrelem 15859

Description: Lemma for fsumre 15860, fsumim 15861, and fsumcj 15862. (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 11198	. . . . . . . 8 ⊢ 0 ∈ ℂ
2		fsumrelem.3	. . . . . . . . 9 ⊢ 𝐹:ℂ⟶ℂ
3	2	ffvelcdmi 7079	. . . . . . . 8 ⊢ (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
4	1, 3	ax-mp 5	. . . . . . 7 ⊢ (𝐹‘0) ∈ ℂ
5	4	addridi 11397	. . . . . 6 ⊢ ((𝐹‘0) + 0) = (𝐹‘0)
6		fvoveq1 7434	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘0))
8	7	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘𝑦)))
9	6, 8	eqeq12d 2785	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹‘𝑦))))
10		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11		00id 11385	. . . . . . . . . . 11 ⊢ (0 + 0) = 0
12	10, 11	eqtrdi 2820	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (0 + 𝑦) = 0)
13	12	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
15	14	oveq2d 7427	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝐹‘0) + (𝐹‘𝑦)) = ((𝐹‘0) + (𝐹‘0)))
16	13, 15	eqeq12d 2785	. . . . . . . 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 3549	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
19	1, 1, 18	mp2an 704	. . . . . 6 ⊢ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
20	5, 19	eqtr2i 2793	. . . . 5 ⊢ ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
21	4, 4, 1	addcani 11403	. . . . 5 ⊢ (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
22	20, 21	mpbi 233	. . . 4 ⊢ (𝐹‘0) = 0
23		sumeq1 15740	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24		sum0 15772	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
25	23, 24	eqtrdi 2820	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
26	25	fveq2d 6886	. . . 4 ⊢ (𝐴 = ∅ → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = (𝐹‘0))
27		sumeq1 15740	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = Σ𝑘 ∈ ∅ (𝐹‘𝐵))
28		sum0 15772	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐹‘𝐵) = 0
29	27, 28	eqtrdi 2820	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐹‘𝐵) = 0)
30	22, 26, 29	3eqtr4a 2830	. . 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 11182	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
33	32	adantl 486	. . . . . . . 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 7111	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
36	35	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
37		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
38		f1of 6821	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
39	37, 38	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
40		fco 6731	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
41	36, 39, 40	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
42	41	ffvelcdmda 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43		simprl 782	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
44		nnuz 12901	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
45	43, 44	eleqtrdi 2879	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
46	17	adantl 486	. . . . . . . 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 7080	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑥) ∈ 𝐴)
48		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
49		eqid 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
50	49	fvmpt2 7002	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
51	48, 34, 50	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
52	51	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘𝐵))
53		fvex 6895	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝐵) ∈ V
54		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))
55	54	fvmpt2 7002	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ V) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
56	48, 53, 55	sylancl 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = (𝐹‘𝐵))
57	52, 56	eqtr4d 2807	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
58	57	ralrimiva 3163	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘))
59	58	ad2antrr 738	. . . . . . . . . 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 2931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐹
61		nffvmpt1 6893	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))
62	60, 61	nffv 6892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
63		nffvmpt1 6893	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
64	62, 63	nfeq 2944	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥))
65		2fveq3 6887	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐹‘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥))))
66		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
67	65, 66	eqeq12d 2785	. . . . . . . . . . 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 3576	. . . . . . . . . 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 66	. . . . . . . . 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 6982	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
71	39, 70	sylan 591	. . . . . . . . . 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 6886	. . . . . . . . 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 6982	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
74	39, 73	sylan 591	. . . . . . . . 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 2814	. . . . . . . 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 14085	. . . . . . 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 6882	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑥)))
78	36	ffvelcdmda 7080	. . . . . . . . 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 15771	. . . . . . . 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 6886	. . . . . . 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 6882	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑥) → ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘(𝑓‘𝑥)))
82	2	ffvelcdmi 7079	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℂ → (𝐹‘𝐵) ∈ ℂ)
83	34, 82	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℂ)
84	83	fmpttd 7111	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
85	84	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)):𝐴⟶ℂ)
86	85	ffvelcdmda 7080	. . . . . . . 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 15771	. . . . . . 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 2814	. . . . . 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 15760	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
90	89	fveq2i 6885	. . . . . 6 ⊢ (𝐹‘Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐹‘Σ𝑘 ∈ 𝐴 𝐵)
91		sumfc 15760	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)
92	88, 90, 91	3eqtr3g 2827	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))
93	92	expr 461	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
94	93	exlimdv 1960	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵)))
95	94	expimpd 458	. 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 15761	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
98	96, 97	syl 18	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
99	31, 95, 98	mpjaod 873	1 ⊢ (𝜑 → (𝐹‘Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐹‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  Vcvv 3461  ∅c0 4292   ↦ cmpt 5194   ∘ ccom 5666  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  Fincfn 8943  ℂcc 11098  0cc0 11100  1c1 11101   + caddc 11103  ℕcn 12233  ℤ_≥cuz 12862  ...cfz 13535  seqcseq 14037  ♯chash 14366  Σcsu 15737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-fzo 13683  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-sum 15738

This theorem is referenced by:  fsumre  15860  fsumim  15861  fsumcj  15862