Theorem fsummulc2 15691

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 11315	. . 3 ⊢ (𝜑 → (𝐶 · 0) = 0)
3		sumeq1 15596	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4		sum0 15628	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
5	3, 4	eqtrdi 2780	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
6	5	oveq2d 7365	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0))
7		sumeq1 15596	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8		sum0 15628	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
9	7, 8	eqtrdi 2780	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0)
10	6, 9	eqeq12d 2745	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
11	2, 10	syl5ibrcom 247	. 2 ⊢ (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
12		addcl 11091	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝑛 + 𝑚) ∈ ℂ)
13	12	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑛 + 𝑚) ∈ ℂ)
14	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐶 ∈ ℂ)
15		adddi 11098	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ) → (𝐶 · (𝑛 + 𝑚)) = ((𝐶 · 𝑛) + (𝐶 · 𝑚)))
16	15	3expb 1120	. . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
19		nnuz 12778	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
20	18, 19	eleqtrdi 2838	. . . . . . . 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 7049	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
23	22	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
24		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
25	24	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
26		f1of 6764	. . . . . . . . . . 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 6676	. . . . . . . . . 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 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ (1...(♯‘𝐴)))
31	29, 30	ffvelcdmd 7019	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
32	27, 30	ffvelcdmd 7019	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑛) ∈ 𝐴)
33		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
34	1	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
35	34, 21	mulcld 11135	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
36		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))
37	36	fvmpt2 6941	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
40	39	fvmpt2 6941	. . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐶 · 𝐵))
43	38, 42	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)))
44	43	ralrimiva 3121	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)))
45	44	ad2antrr 726	. . . . . . . . . 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 6833	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛))
47		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐶
48		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
49		nffvmpt1 6833	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
50	47, 48, 49	nfov 7379	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
51	46, 50	nfeq 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
52		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
53		fveq2 6822	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
54	53	oveq2d 7365	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑛) → (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘)) = (𝐶 · ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))))
55	52, 54	eqeq12d 2745	. . . . . . . . . . 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 3565	. . . . . . . . . 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 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
59		fvco3 6922	. . . . . . . . . 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 6922	. . . . . . . . . . 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 7365	. . . . . . . . 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 2774	. . . . . . . 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 13960	. . . . . . 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 6822	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
67	35	fmpttd 7049	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
68	67	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
69	68	ffvelcdmda 7018	. . . . . . . 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 15627	. . . . . . 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 6822	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
72	22	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
73	72	ffvelcdmda 7018	. . . . . . . . 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 15627	. . . . . . . 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 7365	. . . . . . 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 2775	. . . . . 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 15616	. . . . . . 7 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵
78	77	oveq2i 7360	. . . . . 6 ⊢ (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵)
79		sumfc 15616	. . . . . 6 ⊢ Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)
80	76, 78, 79	3eqtr3g 2787	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
81	80	expr 456	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
82	81	exlimdv 1933	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
83	82	expimpd 453	. 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 15617	. . 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 860	1 ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  ∅c0 4284   ↦ cmpt 5173   ∘ ccom 5623  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  ℂcc 11007  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  ℕcn 12128  ℤ_≥cuz 12735  ...cfz 13410  seqcseq 13908  ♯chash 14237  Σcsu 15593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-oi 9402  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-sum 15594

This theorem is referenced by:  fsummulc1  15692  fsumneg  15694  fsum2mul  15696  incexc2  15745  pwdif  15775  mertens  15793  binomrisefac  15949  fsumkthpow  15963  eirrlem  16113  pwp1fsum  16302  csbren  25297  trirn  25298  itg1addlem4  25598  itg1addlem5  25599  itg1mulc  25603  elqaalem3  26227  advlogexp  26562  fsumharmonic  26920  basellem8  26996  muinv  27101  fsumdvdsmul  27103  fsumdvdsmulOLD  27105  logfaclbnd  27131  dchrsum2  27177  sumdchr2  27179  rplogsumlem2  27394  rpvmasumlem  27396  dchrmusum2  27403  dchrvmasumlem1  27404  dchrvmasum2lem  27405  dchrvmasumlem2  27407  dchrvmasumiflem1  27410  rpvmasum2  27421  dchrisum0lem2  27427  mudivsum  27439  mulogsum  27441  mulog2sumlem1  27443  mulog2sumlem2  27444  mulog2sumlem3  27445  vmalogdivsum2  27447  logsqvma  27451  selberglem1  27454  selberglem2  27455  selberg  27457  selberg3lem1  27466  selberg4lem1  27469  selberg4  27470  selbergr  27477  selberg3r  27478  selberg34r  27480  pntsval2  27485  pntrlog2bndlem2  27487  pntrlog2bndlem3  27488  pntrlog2bndlem4  27489  pntrlog2bndlem6  27492  pntpbnd2  27496  pntlemk  27515  axsegconlem9  28870  ax5seglem1  28873  ax5seglem2  28874  ax5seglem9  28882  hgt750lemf  34621  hgt750lemb  34624  knoppndvlem11  36496  3factsumint4  41997  lcmineqlem6  42007  oddnumth  42284  jm2.22  42968  dvnprodlem2  45928  stoweidlem26  46007  stirlinglem12  46066  fourierdlem83  46170  etransclem46  46261  altgsumbcALT  48337  aacllem  49786