fsummulc2 - Intuitionistic Logic Explorer

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Theorem fsummulc2 12134

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 8666	. . 3 ⊢ (𝜑 → (𝐶 · 0) = 0)
3		sumeq1 12040	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4		sum0 12074	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
5	3, 4	eqtrdi 2281	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
6	5	oveq2d 6066	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0))
7		sumeq1 12040	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8		sum0 12074	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
9	7, 8	eqtrdi 2281	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0)
10	6, 9	eqeq12d 2247	. . 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 8252	. . . . . . . . 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 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
16		simprr 533	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ)
17	14, 15, 16	adddid 8298	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣)))
18		simprl 531	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
19		nnuz 9890	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
20	18, 19	eleqtrdi 2325	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
21		elnnuz 9891	. . . . . . . . . . . 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 5614	. . . . . . . . . . . . . . 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 9604	. . . . . . . . . . . . . . 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 9699	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
30		eluzelz 9863	. . . . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ))
33		eluzle 9866	. . . . . . . . . . . . . . . 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 10349	. . . . . . . . . . . . . 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 5748	. . . . . . . . . . . . 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 5813	. . . . . . . . . . . . . 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 2615	. . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵
46	45	nfel1 2395	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ
47		csbeq1a 3147	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑢) → 𝐵 = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
48	47	eleq1d 2301	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑢) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ))
49	46, 48	rspc 2915	. . . . . . . . . . . . . . 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 2232	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
52	51	fvmpts 5755	. . . . . . . . . . . . . 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 2309	. . . . . . . . . . . 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 2309	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ)
56		0cnd 8267	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
57	23	nnzd 9699	. . . . . . . . . . . 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 9699	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
60		zdcle 9654	. . . . . . . . . . . 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 3652	. . . . . . . . . 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 4112	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
64		fveq2 5670	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢))
65	63, 64	ifbieq1d 3645	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))
66		eqid 2232	. . . . . . . . . . 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 5753	. . . . . . . . . 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 2309	. . . . . . . 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 6092	. . . . . . . . . . . 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 3629	. . . . . . . . . . . 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 5748	. . . . . . . . . . . . 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 8294	. . . . . . . . . . . . . 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 2309	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ)
78		eqid 2232	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))
79	78	fvmpts 5755	. . . . . . . . . . . . 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 2269	. . . . . . . . . . 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 3629	. . . . . . . . . . . . 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 2269	. . . . . . . . . . . 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 6066	. . . . . . . . . . 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 2275	. . . . . . . . . 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 8666	. . . . . . . . . . 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 3632	. . . . . . . . . . . 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 6066	. . . . . . . . . . 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 3632	. . . . . . . . . . 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 2276	. . . . . . . . . 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 844	. . . . . . . . . . 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 806	. . . . . . . . 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 2309	. . . . . . . . . . . 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 2309	. . . . . . . . . . 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 3652	. . . . . . . . . 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 5670	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
100	63, 99	ifbieq1d 3645	. . . . . . . . . . 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 2232	. . . . . . . . . . 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 5753	. . . . . . . . . 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 6066	. . . . . . . . 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 2275	. . . . . . . 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 8254	. . . . . . . . 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 10894	. . . . . . 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 5670	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
111		simprr 533	. . . . . . . 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 8294	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
114	113	fmpttd 5832	. . . . . . . . . 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 5812	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
117		fvco3 5748	. . . . . . . . 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 12073	. . . . . . 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 5670	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
121	42	fmpttd 5832	. . . . . . . . . . 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 5812	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
124		fvco3 5748	. . . . . . . . . 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 12073	. . . . . . . 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 6066	. . . . . . 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 2276	. . . . . 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 12059	. . . . . . . . 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 6066	. . . . . . 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 2615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 · 𝐵) ∈ ℂ)
134		sumfct 12059	. . . . . . . 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 2273	. . . . 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 1868	. . 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 12060	. . 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 726	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 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2203 ∀wral 2520 ⦋csb 3138 ∅c0 3508 ifcif 3620 class class class wbr 4109 ↦ cmpt 4171 ∘ ccom 4753 ⟶wf 5348 –1-1-onto→wf1o 5351 ‘cfv 5352 (class class class)co 6050 Fincfn 6975 ℂcc 8125 0cc0 8127 1c1 8128 + caddc 8130 · cmul 8132 ≤ cle 8309 ℕcn 9237 ℤcz 9577 ℤ_≥cuz 9853 ...cfz 10342 seqcseq 10809 ♯chash 11138 Σcsu 12038

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-isom 5361 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-irdg 6601 df-frec 6622 df-1o 6647 df-oadd 6651 df-er 6767 df-en 6976 df-dom 6977 df-fin 6978 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-fz 10343 df-fzo 10477 df-seqfrec 10810 df-exp 10901 df-ihash 11139 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-clim 11964 df-sumdc 12039

This theorem is referenced by: fsummulc1 12135 fsumneg 12137 fsum2mul 12139 cvgratnnlemabsle 12213 mertensabs 12223 eirraplem 12463 fsumdvdsmul 15859

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