fsummulc2 - Intuitionistic Logic Explorer

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Theorem fsummulc2 11340

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 8263	. . 3 ⊢ (𝜑 → (𝐶 · 0) = 0)
3		sumeq1 11247	. . . . . 6 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
4		sum0 11280	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
5	3, 4	eqtrdi 2206	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = 0)
6	5	oveq2d 5837	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = (𝐶 · 0))
7		sumeq1 11247	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = Σ𝑘 ∈ ∅ (𝐶 · 𝐵))
8		sum0 11280	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐶 · 𝐵) = 0
9	7, 8	eqtrdi 2206	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) = 0)
10	6, 9	eqeq12d 2172	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵) ↔ (𝐶 · 0) = 0))
11	2, 10	syl5ibrcom 156	. 2 ⊢ (𝜑 → (𝐴 = ∅ → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
12		addcl 7852	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ)
13	12	adantl 275	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ)
14	1	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝐶 ∈ ℂ)
15		simprl 521	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑢 ∈ ℂ)
16		simprr 522	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → 𝑣 ∈ ℂ)
17	14, 15, 16	adddid 7897	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝐶 · (𝑢 + 𝑣)) = ((𝐶 · 𝑢) + (𝐶 · 𝑣)))
18		simprl 521	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
19		nnuz 9469	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
20	18, 19	eleqtrdi 2250	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
21		elnnuz 9470	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℕ ↔ 𝑢 ∈ (ℤ_≥‘1))
22	21	biimpri 132	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℕ)
23	22	adantl 275	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℕ)
24		f1of 5413	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
25	24	ad2antll 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
26	25	ad2antrr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
27		1zzd 9189	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
28	18	ad2antrr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
29	28	nnzd 9280	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
30		eluzelz 9443	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℤ)
31	30	ad2antlr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ ℤ)
32	27, 29, 31	3jca 1162	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ))
33		eluzle 9446	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 1 ≤ 𝑢)
34	33	ad2antlr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 1 ≤ 𝑢)
35		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ≤ (♯‘𝐴))
36	34, 35	jca 304	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴)))
37		elfz2 9914	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑢 ∈ ℤ) ∧ (1 ≤ 𝑢 ∧ 𝑢 ≤ (♯‘𝐴))))
38	32, 36, 37	sylanbrc 414	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝑢 ∈ (1...(♯‘𝐴)))
39		fvco3 5538	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑢 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)))
40	26, 38, 39	syl2anc 409	. . . . . . . . . . . 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	ffvelrnd 5602	. . . . . . . . . . . . . 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 2530	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
44	43	ad3antrrr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
45		nfcsb1v 3064	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵
46	45	nfel1 2310	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ
47		csbeq1a 3040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑢) → 𝐵 = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
48	47	eleq1d 2226	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑢) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ))
49	46, 48	rspc 2810	. . . . . . . . . . . . . . 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 2157	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
52	51	fvmpts 5545	. . . . . . . . . . . . . 14 ⊢ (((𝑓‘𝑢) ∈ 𝐴 ∧ ⦋(𝑓‘𝑢) / 𝑘⦌𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑢)) = ⦋(𝑓‘𝑢) / 𝑘⦌𝐵)
53	41, 50, 52	syl2anc 409	. . . . . . . . . . . . 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 2234	. . . . . . . . . . . 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 2234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢) ∈ ℂ)
56		0cnd 7866	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
57	23	nnzd 9280	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℤ)
58	18	adantr 274	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℕ)
59	58	nnzd 9280	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
60		zdcle 9235	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑢 ≤ (♯‘𝐴))
61	57, 59, 60	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ≤ (♯‘𝐴))
62	55, 56, 61	ifcldadc 3534	. . . . . . . . . 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 3968	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑢 ≤ (♯‘𝐴)))
64		fveq2 5467	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢))
65	63, 64	ifbieq1d 3527	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑢 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑢 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑢), 0))
66		eqid 2157	. . . . . . . . . . 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 5543	. . . . . . . . . 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 409	. . . . . . . . 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 2234	. . . . . . . 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 5859	. . . . . . . . . . . 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 3512	. . . . . . . . . . . 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 5538	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
76	75, 50	mulcld 7893	. . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑢) / 𝑘⦌(𝐶 · 𝐵) ∈ ℂ)
78		eqid 2157	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))
79	78	fvmpts 5545	. . . . . . . . . . . . 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 409	. . . . . . . . . . . 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 2194	. . . . . . . . . . 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 3512	. . . . . . . . . . . . 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 2194	. . . . . . . . . . . 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 5837	. . . . . . . . . . 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 2200	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → 𝐶 ∈ ℂ)
87	86	mul01d 8263	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → (𝐶 · 0) = 0)
88		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ≤ (♯‘𝐴)) → ¬ 𝑢 ≤ (♯‘𝐴))
89	88	iffalsed 3515	. . . . . . . . . . . 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 5837	. . . . . . . . . . 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 3515	. . . . . . . . . . 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 2201	. . . . . . . . . 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 822	. . . . . . . . . . 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 788	. . . . . . . . 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 2234	. . . . . . . . . . . 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 2234	. . . . . . . . . . 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 3534	. . . . . . . . . 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 5467	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑢 → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑢))
100	63, 99	ifbieq1d 3527	. . . . . . . . . . 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 2157	. . . . . . . . . . 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 5543	. . . . . . . . . 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 409	. . . . . . . . 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 5837	. . . . . . . . 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 2200	. . . . . . . 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 7854	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 · 𝑣) ∈ ℂ)
107	106	adantl 275	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 · 𝑣) ∈ ℂ)
108	1	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝐶 ∈ ℂ)
109	13, 17, 20, 69, 105, 107, 108	seq3distr 10407	. . . . . . 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 5467	. . . . . . . 8 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
111		simprr 522	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
112	1	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
113	112, 42	mulcld 7893	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 · 𝐵) ∈ ℂ)
114	113	fmpttd 5621	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
115	114	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)):𝐴⟶ℂ)
116	115	ffvelrnda 5601	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘𝑚) ∈ ℂ)
117		fvco3 5538	. . . . . . . . 9 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐶 · 𝐵))‘(𝑓‘𝑛)))
118	25, 117	sylan 281	. . . . . . . 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 11279	. . . . . . 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 5467	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
121	42	fmpttd 5621	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
122	121	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
123	122	ffvelrnda 5601	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
124		fvco3 5538	. . . . . . . . . 10 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
125	25, 124	sylan 281	. . . . . . . . 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 11279	. . . . . . . 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 5837	. . . . . . 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 2201	. . . . . 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 11266	. . . . . . . . 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 5837	. . . . . . 7 ⊢ (𝜑 → (𝐶 · Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚)) = (𝐶 · Σ𝑘 ∈ 𝐴 𝐵))
132	131	adantr 274	. . . . . 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 2530	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐶 · 𝐵) ∈ ℂ)
134		sumfct 11266	. . . . . . . 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 274	. . . . . 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 2198	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))
138	137	expr 373	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
139	138	exlimdv 1799	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵)))
140	139	expimpd 361	. 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 11267	. . 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 708	1 ⊢ (𝜑 → (𝐶 · Σ𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (𝐶 · 𝐵))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 820 ∧ w3a 963 = wceq 1335 ∃wex 1472 ∈ wcel 2128 ∀wral 2435 ⦋csb 3031 ∅c0 3394 ifcif 3505 class class class wbr 3965 ↦ cmpt 4025 ∘ ccom 4589 ⟶wf 5165 –1-1-onto→wf1o 5168 ‘cfv 5169 (class class class)co 5821 Fincfn 6682 ℂcc 7725 0cc0 7727 1c1 7728 + caddc 7730 · cmul 7732 ≤ cle 7908 ℕcn 8828 ℤcz 9162 ℤ_≥cuz 9434 ...cfz 9907 seqcseq 10339 ♯chash 10644 Σcsu 11245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-mulrcl 7826 ax-addcom 7827 ax-mulcom 7828 ax-addass 7829 ax-mulass 7830 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-1rid 7834 ax-0id 7835 ax-rnegex 7836 ax-precex 7837 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-apti 7842 ax-pre-ltadd 7843 ax-pre-mulgt0 7844 ax-pre-mulext 7845 ax-arch 7846 ax-caucvg 7847

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-en 6683 df-dom 6684 df-fin 6685 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-reap 8445 df-ap 8452 df-div 8541 df-inn 8829 df-2 8887 df-3 8888 df-4 8889 df-n0 9086 df-z 9163 df-uz 9435 df-q 9524 df-rp 9556 df-fz 9908 df-fzo 10037 df-seqfrec 10340 df-exp 10414 df-ihash 10645 df-cj 10737 df-re 10738 df-im 10739 df-rsqrt 10893 df-abs 10894 df-clim 11171 df-sumdc 11246

This theorem is referenced by: fsummulc1 11341 fsumneg 11343 fsum2mul 11345 cvgratnnlemabsle 11419 mertensabs 11429 eirraplem 11668

W3C validator