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Theorem fsumadd 11414

Description: The sum of two finite sums. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 22-Apr-2014.)

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

**Assertion**
Ref	Expression
fsumadd	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))

Distinct variable groups: 𝐴,𝑘 𝜑,𝑘 Allowed substitution hints: 𝐵(𝑘) 𝐶(𝑘)

Proof of Theorem fsumadd Dummy variables 𝑓 𝑗 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		00id 8098	. . . . 5 ⊢ (0 + 0) = 0
2		sum0 11396	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐵 = 0
3		sum0 11396	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ 𝐶 = 0
4	2, 3	oveq12i 5887	. . . . 5 ⊢ (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶) = (0 + 0)
5		sum0 11396	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = 0
6	1, 4, 5	3eqtr4ri 2209	. . . 4 ⊢ Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶)
7		sumeq1 11363	. . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = Σ𝑘 ∈ ∅ (𝐵 + 𝐶))
8		sumeq1 11363	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
9		sumeq1 11363	. . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
10	8, 9	oveq12d 5893	. . . 4 ⊢ (𝐴 = ∅ → (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶))
11	6, 7, 10	3eqtr4a 2236	. . 3 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))
12	11	a1i 9	. 2 ⊢ (𝜑 → (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)))
13		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
14		nnuz 9563	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
15	13, 14	eleqtrdi 2270	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (♯‘𝐴) ∈ (ℤ_≥‘1))
16		eqid 2177	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))
17		breq1 4007	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑛 ≤ (♯‘𝐴)))
18		fveq2 5516	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛))
19	17, 18	ifbieq1d 3557	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))
20		elnnuz 9564	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ_≥‘1))
21	20	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘1) → 𝑛 ∈ ℕ)
22	21	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℕ)
23		fsumadd.2	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
24	23	adantlr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
25	24	fmpttd 5672	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
26		simprr 531	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
27		f1of 5462	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
28	26, 27	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
29		fco 5382	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
30	25, 28, 29	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
31	30	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
32		1zzd 9280	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
33	13	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
34	33	nnzd 9374	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
35		eluzelz 9537	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘1) → 𝑛 ∈ ℤ)
36	35	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ ℤ)
37	32, 34, 36	3jca 1177	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ))
38		eluzle 9540	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘1) → 1 ≤ 𝑛)
39	38	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ≤ 𝑛)
40		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ≤ (♯‘𝐴))
41	39, 40	jca 306	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ≤ 𝑛 ∧ 𝑛 ≤ (♯‘𝐴)))
42		elfz2 10015	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (1 ≤ 𝑛 ∧ 𝑛 ≤ (♯‘𝐴))))
43	37, 41, 42	sylanbrc 417	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ (1...(♯‘𝐴)))
44	31, 43	ffvelcdmd 5653	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
45		0cnd 7950	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
46	22	nnzd 9374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → 𝑛 ∈ ℤ)
47	13	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℕ)
48	47	nnzd 9374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
49		zdcle 9329	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑛 ≤ (♯‘𝐴))
50	46, 48, 49	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → DECID 𝑛 ≤ (♯‘𝐴))
51	44, 45, 50	ifcldadc 3564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
52	16, 19, 22, 51	fvmptd3 5610	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))
53	52, 51	eqeltrd 2254	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
54		eqid 2177	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))
55		fveq2 5516	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛))
56	17, 55	ifbieq1d 3557	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0))
57		fsumadd.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
58	57	adantlr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
59	58	fmpttd 5672	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
60	59	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ)
61	28	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
62		fco 5382	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
63	60, 61, 62	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
64	63, 43	ffvelcdmd 5653	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
65	64, 45, 50	ifcldadc 3564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
66	54, 56, 22, 65	fvmptd3 5610	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0))
67	66, 65	eqeltrd 2254	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
68		simpll 527	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
69	28	ffvelcdmda 5652	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓‘𝑛) ∈ 𝐴)
70		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
71	23, 57	addcld 7977	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ)
72		eqid 2177	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))
73	72	fvmpt2 5600	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ 𝐴 ∧ (𝐵 + 𝐶) ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
74	70, 71, 73	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
75		eqid 2177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)
76	75	fvmpt2 5600	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
77	70, 23, 76	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = 𝐵)
78		eqid 2177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
79	78	fvmpt2 5600	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
80	70, 57, 79	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
81	77, 80	oveq12d 5893	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) = (𝐵 + 𝐶))
82	74, 81	eqtr4d 2213	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
83	82	ralrimiva 2550	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
84	83	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)))
85		nffvmpt1 5527	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛))
86		nffvmpt1 5527	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛))
87		nfcv 2319	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 +
88		nffvmpt1 5527	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))
89	86, 87, 88	nfov 5905	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
90	85, 89	nfeq 2327	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
91		fveq2 5516	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)))
92		fveq2 5516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
93		fveq2 5516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
94	92, 93	oveq12d 5893	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑓‘𝑛) → (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
95	91, 94	eqeq12d 2192	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) ↔ ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))))
96	90, 95	rspc 2836	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑘) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))))
97	69, 84, 96	sylc 62	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
98		fvco3 5588	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)))
99	28, 98	sylan 283	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)))
100		fvco3 5588	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
101	28, 100	sylan 283	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
102		fvco3 5588	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...(♯‘𝐴))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
103	28, 102	sylan 283	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
104	101, 103	oveq12d 5893	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) + (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)) + ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛))))
105	97, 99, 104	3eqtr4d 2220	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) + (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)))
106	68, 43, 105	syl2anc 411	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) + (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)))
107	40	iftrued 3542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛))
108	40	iftrued 3542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛))
109	40	iftrued 3542	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛))
110	108, 109	oveq12d 5893	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)) = ((((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) + (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛)))
111	106, 107, 110	3eqtr4d 2220	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))
112	1	eqcomi 2181	. . . . . . . . . . 11 ⊢ 0 = (0 + 0)
113		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → ¬ 𝑛 ≤ (♯‘𝐴))
114	113	iffalsed 3545	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = 0)
115	113	iffalsed 3545	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = 0)
116	113	iffalsed 3545	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0) = 0)
117	115, 116	oveq12d 5893	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)) = (0 + 0))
118	112, 114, 117	3eqtr4a 2236	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))
119		exmiddc 836	. . . . . . . . . . 11 ⊢ (DECID 𝑛 ≤ (♯‘𝐴) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
120	50, 119	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
121	111, 118, 120	mpjaodan 798	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))
122		eqid 2177	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
123		fveq2 5516	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑛 → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗) = (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛))
124	17, 123	ifbieq1d 3557	. . . . . . . . . 10 ⊢ (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
125	71	fmpttd 5672	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
126	125	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
127		fco 5382	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
128	126, 61, 127	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
129	128, 43	ffvelcdmd 5653	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) ∈ ℂ)
130	129, 45, 50	ifcldadc 3564	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
131	122, 124, 22, 130	fvmptd3 5610	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
132	52, 66	oveq12d 5893	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → (((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) + ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛)) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))
133	121, 131, 132	3eqtr4d 2220	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (ℤ_≥‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))‘𝑛) = (((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) + ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛)))
134	15, 53, 67, 133	ser3add 10505	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)) = ((seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))))
135		fveq2 5516	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘(𝑓‘𝑛)))
136	24, 58	addcld 7977	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℂ)
137	136	fmpttd 5672	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
138	137	ffvelcdmda 5652	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) ∈ ℂ)
139	135, 13, 26, 138, 99	fsum3 11395	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
140		breq1 4007	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
141		fveq2 5516	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗))
142	140, 141	ifbieq1d 3557	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
143	142	cbvmptv 4100	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
144		seqeq3 10450	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))))
145	143, 144	ax-mp 5	. . . . . . . . 9 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))
146	145	fveq1i 5517	. . . . . . . 8 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
147	139, 146	eqtrdi 2226	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)))
148		fveq2 5516	. . . . . . . . . 10 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝑓‘𝑛)))
149	25	ffvelcdmda 5652	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ)
150	148, 13, 26, 149, 101	fsum3 11395	. . . . . . . . 9 ⊢ ((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)))
151		fveq2 5516	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗))
152	140, 151	ifbieq1d 3557	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))
153	152	cbvmptv 4100	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))
154		seqeq3 10450	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0))))
155	153, 154	ax-mp 5	. . . . . . . . . 10 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)))
156	155	fveq1i 5517	. . . . . . . . 9 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
157	150, 156	eqtrdi 2226	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)))
158		fveq2 5516	. . . . . . . . . 10 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘(𝑓‘𝑛)))
159	59	ffvelcdmda 5652	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) ∈ ℂ)
160	158, 13, 26, 159, 103	fsum3 11395	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
161		fveq2 5516	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑗 → (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛) = (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗))
162	140, 161	ifbieq1d 3557	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))
163	162	cbvmptv 4100	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))
164		seqeq3 10450	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0))))
165	163, 164	ax-mp 5	. . . . . . . . . 10 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)))
166	165	fveq1i 5517	. . . . . . . . 9 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
167	160, 166	eqtrdi 2226	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)))
168	157, 167	oveq12d 5893	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) + Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = ((seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘 ∈ 𝐴 ↦ 𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))))
169	134, 147, 168	3eqtr4d 2220	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) + Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)))
170	71	ralrimiva 2550	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐵 + 𝐶) ∈ ℂ)
171		sumfct 11382	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 (𝐵 + 𝐶) ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶))
172	170, 171	syl 14	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶))
173	172	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶))
174	23	ralrimiva 2550	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
175		sumfct 11382	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
176	174, 175	syl 14	. . . . . . . 8 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐵)
177	57	ralrimiva 2550	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
178		sumfct 11382	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
179	177, 178	syl 14	. . . . . . . 8 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
180	176, 179	oveq12d 5893	. . . . . . 7 ⊢ (𝜑 → (Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) + Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))
181	180	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → (Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) + Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))
182	169, 173, 181	3eqtr3d 2218	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))
183	182	expr 375	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)))
184	183	exlimdv 1819	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)))
185	184	expimpd 363	. 2 ⊢ (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶)))
186		fsumadd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
187		fz1f1o 11383	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
188	186, 187	syl 14	. 2 ⊢ (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
189	12, 185, 188	mpjaod 718	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐵 + 𝐶) = (Σ𝑘 ∈ 𝐴 𝐵 + Σ𝑘 ∈ 𝐴 𝐶))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 ∧ w3a 978 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 ∅c0 3423 ifcif 3535 class class class wbr 4004 ↦ cmpt 4065 ∘ ccom 4631 ⟶wf 5213 –1-1-onto→wf1o 5216 ‘cfv 5217 (class class class)co 5875 Fincfn 6740 ℂcc 7809 0cc0 7811 1c1 7812 + caddc 7814 ≤ cle 7993 ℕcn 8919 ℤcz 9253 ℤ_≥cuz 9528 ...cfz 10008 seqcseq 10445 ♯chash 10755 Σcsu 11361

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-en 6741 df-dom 6742 df-fin 6743 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-fz 10009 df-fzo 10143 df-seqfrec 10446 df-exp 10520 df-ihash 10756 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-sumdc 11362

This theorem is referenced by: fsumsplit 11415 fsumsub 11460 binomlem 11491 pcbc 12349

W3C validator