fsummulc2 - Intuitionistic Logic Explorer

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Theorem fsummulc2 11473

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 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsummulc2.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ)
2	1	mul01d 8367	. . 3 ⊢ (𝜑 → (𝐶 · 0) = 0)
3		sumeq1 11380	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4		sum0 11413	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
5	3, 4	eqtrdi 2237	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
6	5	oveq2d 5906	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0))
7		sumeq1 11380	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8		sum0 11413	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
9	7, 8	eqtrdi 2237	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0)
10	6, 9	eqeq12d 2203	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
11	2, 10	syl5ibrcom 157	. 2 ⊢ (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
12		addcl 7953	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ)
13	12	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ)
14	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝐶 ∈ ℂ)
15		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
16		simprr 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ)
17	14, 15, 16	adddid 7999	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣)))
18		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
19		nnuz 9580	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
20	18, 19	eleqtrdi 2281	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
21		elnnuz 9581	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℕ ↔ 𝑢 ∈ (ℤ_≥‘1))
22	21	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℕ)
23	22	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℕ)
24		f1of 5475	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
25	24	ad2antll 491	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
26	25	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
27		1zzd 9297	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
28	18	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
29	28	nnzd 9391	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
30		eluzelz 9554	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℤ)
31	30	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ ℤ)
32	27, 29, 31	3jca 1178	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ))
33		eluzle 9557	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 1 ≤ 𝑢)
34	33	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ≤ 𝑢)
35		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ≤ (♯‘𝐴))
36	34, 35	jca 306	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴)))
37		elfz2 10032	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ) ∧ (1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴))))
38	32, 36, 37	sylanbrc 417	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ (1...(♯‘𝐴)))
39		fvco3 5602	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)))
40	26, 38, 39	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)))
41	26, 38	ffvelcdmd 5667	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝑓‘𝑢) ∈ 𝐴)
42		fsummulc2.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
43	42	ralrimiva 2562	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
44	43	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
45		nfcsb1v 3104	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵
46	45	nfel1 2342	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ
47		csbeq1a 3080	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑢) → 𝐵 = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
48	47	eleq1d 2257	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑢) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ))
49	46, 48	rspc 2849	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑢) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ))
50	41, 44, 49	sylc 62	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ)
51		eqid 2188	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
52	51	fvmpts 5609	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑢) ∈ 𝐴 ∧ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
53	41, 50, 52	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
54	53, 50	eqeltrd 2265	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) ∈ ℂ)
55	40, 54	eqeltrd 2265	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ)
56		0cnd 7967	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
57	23	nnzd 9391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℤ)
58	18	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℕ)
59	58	nnzd 9391	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
60		zdcle 9346	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑢 ≤ (♯‘𝐴))
61	57, 59, 60	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ≤ (♯‘𝐴))
62	55, 56, 61	ifcldadc 3577	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ)
63		breq1 4020	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
64		fveq2 5529	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢))
65	63, 64	ifbieq1d 3570	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))
66		eqid 2188	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))
67	65, 66	fvmptg 5607	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))
68	23, 62, 67	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))
69	68, 62	eqeltrd 2265	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢) ∈ ℂ)
70		csbov2g 5931	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑢) ∈ 𝐴 → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵))
71	41, 70	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵))
72	35	iftrued 3555	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
73		fvco3 5602	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)))
74	26, 38, 73	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)))
75	1	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
76	75, 50	mulcld 7995	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵) ∈ ℂ)
77	71, 76	eqeltrd 2265	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ)
78		eqid 2188	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))
79	78	fvmpts 5609	. . . . . . . . . . . . 13 ⊢ (((𝑓‘𝑢) ∈ 𝐴 ∧ ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵))
80	41, 77, 79	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵))
81	72, 74, 80	3eqtrd 2225	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵))
82	35	iftrued 3555	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢))
83	82, 40, 53	3eqtrd 2225	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
84	83	oveq2d 5906	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · ⦋(𝑓‘𝑢) / 𝑘⦌𝐵))
85	71, 81, 84	3eqtr4d 2231	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)))
86	1	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
87	86	mul01d 8367	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · 0) = 0)
88		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → ¬ 𝑢 ≤ (♯‘𝐴))
89	88	iffalsed 3558	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0) = 0)
90	89	oveq2d 5906	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)) = (𝐶 · 0))
91	88	iffalsed 3558	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = 0)
92	87, 90, 91	3eqtr4rd 2232	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)))
93		exmiddc 837	. . . . . . . . . . 11 ⊢ (DECID 𝑢 ≤ (♯‘𝐴) → (𝑢 ≤ (♯‘𝐴) ∨ ¬ 𝑢 ≤ (♯‘𝐴)))
94	61, 93	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (𝑢 ≤ (♯‘𝐴) ∨ ¬ 𝑢 ≤ (♯‘𝐴)))
95	85, 92, 94	mpjaodan 799	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)))
96	80, 77	eqeltrd 2265	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑢)) ∈ ℂ)
97	74, 96	eqeltrd 2265	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢) ∈ ℂ)
98	97, 56, 61	ifcldadc 3577	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ)
99		fveq2 5529	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
100	63, 99	ifbieq1d 3570	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
101		eqid 2188	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))
102	100, 101	fvmptg 5607	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℕ ∧ if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0) ∈ ℂ) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
103	23, 98, 102	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢), 0))
104	68	oveq2d 5906	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢)) = (𝐶 · if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0)))
105	95, 103, 104	3eqtr4d 2231	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0))‘𝑢) = (𝐶 · ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))‘𝑢)))
106		mulcl 7955	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
107	106	adantl 277	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ)
108	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐶 ∈ ℂ)
109	13, 17, 20, 69, 105, 107, 108	seq3distr 10530	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))))
110		fveq2 5529	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
111		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
112	1	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
113	112, 42	mulcld 7995	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
114	113	fmpttd 5686	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
115	114	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
116	115	ffvelcdmda 5666	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
117		fvco3 5602	. . . . . . . . 9 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
118	25, 117	sylan 283	. . . . . . . 8 ⊢ (((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)))
119	110, 18, 111, 116, 118	fsum3 11412	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
120		fveq2 5529	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
121	42	fmpttd 5686	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
122	121	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
123	122	ffvelcdmda 5666	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
124		fvco3 5602	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
125	25, 124	sylan 283	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
126	120, 18, 111, 123, 125	fsum3 11412	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
127	126	oveq2d 5906	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴))))
128	109, 119, 127	3eqtr4rd 2232	. . . . . 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))
129		sumfct 11399	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
130	43, 129	syl 14	. . . . . . . 8 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
131	130	oveq2d 5906	. . . . . . 7 ⊢ (𝜑 → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵))
132	131	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵))
133	113	ralrimiva 2562	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 · 𝐵) ∈ ℂ)
134		sumfct 11399	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 (𝐶 · 𝐵) ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
135	133, 134	syl 14	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
136	135	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
137	128, 132, 136	3eqtr3d 2229	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
138	137	expr 375	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
139	138	exlimdv 1829	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
140	139	expimpd 363	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
141		fsummulc2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
142		fz1f1o 11400	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
143	141, 142	syl 14	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
144	11, 140, 143	mpjaod 719	1 ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 DECID wdc 835 ∧ w3a 979 = wceq 1363 ∃wex 1502 ∈ wcel 2159 ∀wral 2467 ⦋csb 3071 ∅c0 3436 ifcif 3548 class class class wbr 4017 ↦ cmpt 4078 ∘ ccom 4644 ⟶wf 5226 –1-1-onto→wf1o 5229 ‘cfv 5230 (class class class)co 5890 Fincfn 6757 ℂcc 7826 0cc0 7828 1c1 7829 + caddc 7831 · cmul 7833 ≤ cle 8010 ℕcn 8936 ℤcz 9270 ℤ_≥cuz 9545 ...cfz 10025 seqcseq 10462 ♯chash 10772 Σcsu 11378

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-mulrcl 7927 ax-addcom 7928 ax-mulcom 7929 ax-addass 7930 ax-mulass 7931 ax-distr 7932 ax-i2m1 7933 ax-0lt1 7934 ax-1rid 7935 ax-0id 7936 ax-rnegex 7937 ax-precex 7938 ax-cnre 7939 ax-pre-ltirr 7940 ax-pre-ltwlin 7941 ax-pre-lttrn 7942 ax-pre-apti 7943 ax-pre-ltadd 7944 ax-pre-mulgt0 7945 ax-pre-mulext 7946 ax-arch 7947 ax-caucvg 7948

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-if 3549 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-po 4310 df-iso 4311 df-iord 4380 df-on 4382 df-ilim 4383 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-isom 5239 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-1st 6158 df-2nd 6159 df-recs 6323 df-irdg 6388 df-frec 6409 df-1o 6434 df-oadd 6438 df-er 6552 df-en 6758 df-dom 6759 df-fin 6760 df-pnf 8011 df-mnf 8012 df-xr 8013 df-ltxr 8014 df-le 8015 df-sub 8147 df-neg 8148 df-reap 8549 df-ap 8556 df-div 8647 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-n0 9194 df-z 9271 df-uz 9546 df-q 9637 df-rp 9671 df-fz 10026 df-fzo 10160 df-seqfrec 10463 df-exp 10537 df-ihash 10773 df-cj 10868 df-re 10869 df-im 10870 df-rsqrt 11024 df-abs 11025 df-clim 11304 df-sumdc 11379

This theorem is referenced by: fsummulc1 11474 fsumneg 11476 fsum2mul 11478 cvgratnnlemabsle 11552 mertensabs 11562 eirraplem 11801

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