Theorem fsummulc2 15494

Description: A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)

Hypotheses

Ref	Expression
fsummulc2.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fsummulc2.2	⊢ (𝜑 → 𝐶 ∈ ℂ)
fsummulc2.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fsummulc2	⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))

Distinct variable groups:   𝐴,𝑘   𝐶,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsummulc2

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

Step	Hyp	Ref	Expression
1		fsummulc2.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
2	1	mul01d 11174	. . 3 ⊢ (𝜑 → (𝐶 · 0) = 0)
3		sumeq1 15398	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4		sum0 15431	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
5	3, 4	eqtrdi 2796	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
6	5	oveq2d 7287	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0))
7		sumeq1 15398	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8		sum0 15431	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
9	7, 8	eqtrdi 2796	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0)
10	6, 9	eqeq12d 2756	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
11	2, 10	syl5ibrcom 246	. 2 ⊢ (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
12		addcl 10954	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) ∈ ℂ)
13	12	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑛 + 𝑚) ∈ ℂ)
14	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐶 ∈ ℂ)
15		adddi 10961	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
16	15	3expb 1119	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
17	14, 16	sylan 580	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
18		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
19		nnuz 12620	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
20	18, 19	eleqtrdi 2851	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ≥‘1))
21		fsummulc2.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
22	21	fmpttd 6986	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
23	22	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
24		simprr 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
25	24	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
26		f1of 6714	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
28		fco 6622	. . . . . . . . . 10 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
29	23, 27, 28	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
30		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ (1...(♯‘𝐴)))
31	29, 30	ffvelrnd 6959	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
32	27, 30	ffvelrnd 6959	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑛) ∈ 𝐴)
33		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
34	1	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
35	34, 21	mulcld 10996	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
36		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))
37	36	fvmpt2 6883	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐶 · 𝐵) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
38	33, 35, 37	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · 𝐵))
39		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
40	39	fvmpt2 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
41	33, 21, 40	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
42	41	oveq2d 7287	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐶 · 𝐵))
43	38, 42	eqtr4d 2783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)))
44	43	ralrimiva 3110	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)))
45	44	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)))
46		nffvmpt1 6782	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛))
47		nfcv 2909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐶
48		nfcv 2909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
49		nffvmpt1 6782	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
50	47, 48, 49	nfov 7301	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
51	46, 50	nfeq 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
52		fveq2 6771	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
53		fveq2 6771	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
54	53	oveq2d 7287	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))))
55	52, 54	eqeq12d 2756	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑛) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) ↔ ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))))
56	51, 55	rspc 3548	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))))
57	32, 45, 56	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))))
58	26	ad2antll 726	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
59		fvco3 6864	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
60	58, 59	sylan 580	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
61		fvco3 6864	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
62	58, 61	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
63	62	oveq2d 7287	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝐶 · (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))))
64	57, 60, 63	3eqtr4d 2790	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (𝐶 · (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛)))
65	13, 17, 20, 31, 64	seqdistr 13772	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)) = (𝐶 · (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))))
66		fveq2 6771	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
67	35	fmpttd 6986	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
68	67	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
69	68	ffvelrnda 6958	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
70	66, 18, 24, 69, 60	fsum 15430	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
71		fveq2 6771	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
72	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
73	72	ffvelrnda 6958	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
74	71, 18, 24, 73, 62	fsum 15430	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴)))
75	74	oveq2d 7287	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · (seq1( + , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓))‘(♯‘𝐴))))
76	65, 70, 75	3eqtr4rd 2791	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚))
77		sumfc 15419	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
78	77	oveq2i 7282	. . . . . 6 ⊢ (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵)
79		sumfc 15419	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)
80	76, 78, 79	3eqtr3g 2803	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
81	80	expr 457	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
82	81	exlimdv 1940	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
83	82	expimpd 454	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
84		fsummulc2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
85		fz1f1o 15420	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
86	84, 85	syl 17	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
87	11, 83, 86	mpjaod 857	1 ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1542  ∃wex 1786   ∈ wcel 2110  ∀wral 3066  ∅c0 4262   ↦ cmpt 5162   ∘ ccom 5594  ⟶wf 6428  –1-1-onto→wf1o 6431  ‘cfv 6432  (class class class)co 7271  Fincfn 8716  ℂcc 10870  0cc0 10872  1c1 10873   + caddc 10875   · cmul 10877  ℕcn 11973  ℤ_≥cuz 12581  ...cfz 13238  seqcseq 13719  ♯chash 14042  Σcsu 15395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-inf2 9377  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-isom 6441  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-sup 9179  df-oi 9247  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  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 12582  df-rp 12730  df-fz 13239  df-fzo 13382  df-seq 13720  df-exp 13781  df-hash 14043  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-clim 15195  df-sum 15396

This theorem is referenced by:  fsummulc1  15495  fsumneg  15497  fsum2mul  15499  incexc2  15548  pwdif  15578  mertens  15596  binomrisefac  15750  fsumkthpow  15764  eirrlem  15911  pwp1fsum  16098  csbren  24561  trirn  24562  itg1addlem4  24861  itg1addlem4OLD  24862  itg1addlem5  24863  itg1mulc  24867  elqaalem3  25479  advlogexp  25808  fsumharmonic  26159  basellem8  26235  muinv  26340  fsumdvdsmul  26342  logfaclbnd  26368  dchrsum2  26414  sumdchr2  26416  rplogsumlem2  26631  rpvmasumlem  26633  dchrmusum2  26640  dchrvmasumlem1  26641  dchrvmasum2lem  26642  dchrvmasumlem2  26644  dchrvmasumiflem1  26647  rpvmasum2  26658  dchrisum0lem2  26664  mudivsum  26676  mulogsum  26678  mulog2sumlem1  26680  mulog2sumlem2  26681  mulog2sumlem3  26682  vmalogdivsum2  26684  logsqvma  26688  selberglem1  26691  selberglem2  26692  selberg  26694  selberg3lem1  26703  selberg4lem1  26706  selberg4  26707  selbergr  26714  selberg3r  26715  selberg34r  26717  pntsval2  26722  pntrlog2bndlem2  26724  pntrlog2bndlem3  26725  pntrlog2bndlem4  26726  pntrlog2bndlem6  26729  pntpbnd2  26733  pntlemk  26752  axsegconlem9  27291  ax5seglem1  27294  ax5seglem2  27295  ax5seglem9  27303  hgt750lemf  32629  hgt750lemb  32632  knoppndvlem11  34698  3factsumint4  40029  lcmineqlem6  40039  jm2.22  40814  dvnprodlem2  43459  stoweidlem26  43538  stirlinglem12  43597  fourierdlem83  43701  etransclem46  43792  altgsumbcALT  45658  aacllem  46474