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Theorem fsumadd 10763
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 (𝜑𝐴 ∈ Fin)
fsumadd.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fsumadd.3 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
Assertion
Ref Expression
fsumadd (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 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.
StepHypRef Expression
1 00id 7602 . . . . 5 (0 + 0) = 0
2 sum0 10744 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
3 sum0 10744 . . . . . 6 Σ𝑘 ∈ ∅ 𝐶 = 0
42, 3oveq12i 5646 . . . . 5 𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶) = (0 + 0)
5 sum0 10744 . . . . 5 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = 0
61, 4, 53eqtr4ri 2119 . . . 4 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶)
7 sumeq1 10708 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = Σ𝑘 ∈ ∅ (𝐵 + 𝐶))
8 sumeq1 10708 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
9 sumeq1 10708 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 5652 . . . 4 (𝐴 = ∅ → (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2146 . . 3 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
1211a1i 9 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
13 simprl 498 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 9023 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14syl6eleq 2180 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 elnnuz 9024 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
1716biimpri 131 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ ℕ)
1817adantl 271 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → 𝑛 ∈ ℕ)
19 fsumadd.2 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2019adantlr 461 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
2120fmpttd 5437 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
22 simprr 499 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
23 f1of 5237 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2422, 23syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
25 fco 5161 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2621, 24, 25syl2anc 403 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2726ad2antrr 472 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
28 1zzd 8747 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
2913ad2antrr 472 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
3029nnzd 8837 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
31 eluzelz 8997 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ ℤ)
3231ad2antlr 473 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ ℤ)
3328, 30, 323jca 1123 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ))
34 eluzle 9000 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → 1 ≤ 𝑛)
3534ad2antlr 473 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ≤ 𝑛)
36 simpr 108 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ≤ (♯‘𝐴))
3735, 36jca 300 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ≤ 𝑛𝑛 ≤ (♯‘𝐴)))
38 elfz2 9400 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (1 ≤ 𝑛𝑛 ≤ (♯‘𝐴))))
3933, 37, 38sylanbrc 408 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ (1...(♯‘𝐴)))
4027, 39ffvelrnd 5419 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
41 0cnd 7460 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
4218nnzd 8837 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → 𝑛 ∈ ℤ)
4313adantr 270 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℕ)
4443nnzd 8837 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
45 zdcle 8793 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑛 ≤ (♯‘𝐴))
4642, 44, 45syl2anc 403 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → DECID 𝑛 ≤ (♯‘𝐴))
4740, 41, 46ifcldadc 3416 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
48 breq1 3840 . . . . . . . . . . . 12 (𝑗 = 𝑛 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑛 ≤ (♯‘𝐴)))
49 fveq2 5289 . . . . . . . . . . . 12 (𝑗 = 𝑛 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛))
5048, 49ifbieq1d 3409 . . . . . . . . . . 11 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
51 eqid 2088 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
5250, 51fvmptg 5364 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) ∈ ℂ) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
5318, 47, 52syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
5453, 47eqeltrd 2164 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
55 fsumadd.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
5655adantlr 461 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
5756fmpttd 5437 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
5857ad2antrr 472 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
5924ad2antrr 472 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
60 fco 5161 . . . . . . . . . . . . 13 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6158, 59, 60syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6261, 39ffvelrnd 5419 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
6362, 41, 46ifcldadc 3416 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
64 fveq2 5289 . . . . . . . . . . . 12 (𝑗 = 𝑛 → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛))
6548, 64ifbieq1d 3409 . . . . . . . . . . 11 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0))
66 eqid 2088 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
6765, 66fvmptg 5364 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) ∈ ℂ) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0))
6818, 63, 67syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0))
6968, 63eqeltrd 2164 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
70 simpll 496 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
7124ffvelrnda 5418 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
72 simpr 108 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑘𝐴)
7319, 55addcld 7486 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
74 eqid 2088 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 ↦ (𝐵 + 𝐶)) = (𝑘𝐴 ↦ (𝐵 + 𝐶))
7574fvmpt2 5370 . . . . . . . . . . . . . . . . . 18 ((𝑘𝐴 ∧ (𝐵 + 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
7672, 73, 75syl2anc 403 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
77 eqid 2088 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
7877fvmpt2 5370 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
7972, 19, 78syl2anc 403 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
80 eqid 2088 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
8180fvmpt2 5370 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
8272, 55, 81syl2anc 403 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
8379, 82oveq12d 5652 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 + 𝐶))
8476, 83eqtr4d 2123 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
8584ralrimiva 2446 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
8685ad2antrr 472 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
87 nffvmpt1 5300 . . . . . . . . . . . . . . . 16 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛))
88 nffvmpt1 5300 . . . . . . . . . . . . . . . . 17 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
89 nfcv 2228 . . . . . . . . . . . . . . . . 17 𝑘 +
90 nffvmpt1 5300 . . . . . . . . . . . . . . . . 17 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
9188, 89, 90nfov 5661 . . . . . . . . . . . . . . . 16 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9287, 91nfeq 2236 . . . . . . . . . . . . . . 15 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
93 fveq2 5289 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
94 fveq2 5289 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
95 fveq2 5289 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9694, 95oveq12d 5652 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
9793, 96eqeq12d 2102 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
9892, 97rspc 2716 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
9971, 86, 98sylc 61 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
100 fvco3 5359 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
10124, 100sylan 277 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
102 fvco3 5359 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
10324, 102sylan 277 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
104 fvco3 5359 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
10524, 104sylan 277 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
106103, 105oveq12d 5652 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
10799, 101, 1063eqtr4d 2130 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
10870, 39, 107syl2anc 403 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
10936iftrued 3396 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛))
11036iftrued 3396 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛))
11136iftrued 3396 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛))
112110, 111oveq12d 5652 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
113108, 109, 1123eqtr4d 2130 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))
1141eqcomi 2092 . . . . . . . . . . 11 0 = (0 + 0)
115 simpr 108 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → ¬ 𝑛 ≤ (♯‘𝐴))
116115iffalsed 3399 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = 0)
117115iffalsed 3399 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = 0)
118115iffalsed 3399 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) = 0)
119117, 118oveq12d 5652 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)) = (0 + 0))
120114, 116, 1193eqtr4a 2146 . . . . . . . . . 10 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))
121 exmiddc 782 . . . . . . . . . . 11 (DECID 𝑛 ≤ (♯‘𝐴) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
12246, 121syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
123113, 120, 122mpjaodan 747 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))
12473fmpttd 5437 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
125124ad3antrrr 476 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
126 fco 5161 . . . . . . . . . . . . 13 (((𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
127125, 59, 126syl2anc 403 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
128127, 39ffvelrnd 5419 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) ∈ ℂ)
129128, 41, 46ifcldadc 3416 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
130 fveq2 5289 . . . . . . . . . . . 12 (𝑗 = 𝑛 → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗) = (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛))
13148, 130ifbieq1d 3409 . . . . . . . . . . 11 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
132 eqid 2088 . . . . . . . . . . 11 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
133131, 132fvmptg 5364 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) ∈ ℂ) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
13418, 129, 133syl2anc 403 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))‘𝑛) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
13553, 68oveq12d 5652 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) + ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛)) = (if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) + if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))
136123, 134, 1353eqtr4d 2130 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))‘𝑛) = (((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) + ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛)))
13715, 54, 69, 136iseradd 9899 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)) = ((seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴))))
138 fveq2 5289 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
13920, 56addcld 7486 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
140139fmpttd 5437 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
141140ffvelrnda 5418 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) ∈ ℂ)
142138, 13, 22, 141, 101fisum 10742 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)))
143 breq1 3840 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
144 fveq2 5289 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗))
145143, 144ifbieq1d 3409 . . . . . . . . . . 11 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
146145cbvmptv 3926 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
147 iseqeq3 9825 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)), ℂ))
148146, 147ax-mp 7 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)), ℂ)
149148fveq1i 5290 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴))
150142, 149syl6eq 2136 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)))
151 fveq2 5289 . . . . . . . . . 10 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
15221ffvelrnda 5418 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
153151, 13, 22, 152, 103fisum 10742 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)))
154 fveq2 5289 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗))
155143, 154ifbieq1d 3409 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
156155cbvmptv 3926 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
157 iseqeq3 9825 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ))
158156, 157ax-mp 7 . . . . . . . . . 10 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ)
159158fveq1i 5290 . . . . . . . . 9 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴))
160153, 159syl6eq 2136 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)))
161 fveq2 5289 . . . . . . . . . 10 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
16257ffvelrnda 5418 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
163161, 13, 22, 162, 105fisum 10742 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)))
164 fveq2 5289 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗))
165143, 164ifbieq1d 3409 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
166165cbvmptv 3926 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
167 iseqeq3 9825 . . . . . . . . . . 11 ((𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ))
168166, 167ax-mp 7 . . . . . . . . . 10 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)), ℂ) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ)
169168fveq1i 5290 . . . . . . . . 9 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)), ℂ)‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴))
170163, 169syl6eq 2136 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)))
171160, 170oveq12d 5652 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = ((seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)), ℂ)‘(♯‘𝐴))))
172137, 150, 1713eqtr4d 2130 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
17373ralrimiva 2446 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 (𝐵 + 𝐶) ∈ ℂ)
174 sumfct 10727 . . . . . . . 8 (∀𝑘𝐴 (𝐵 + 𝐶) ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
175173, 174syl 14 . . . . . . 7 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
176175adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
17719ralrimiva 2446 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
178 sumfct 10727 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
179177, 178syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
18055ralrimiva 2446 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
181 sumfct 10727 . . . . . . . . 9 (∀𝑘𝐴 𝐶 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
182180, 181syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
183179, 182oveq12d 5652 . . . . . . 7 (𝜑 → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
184183adantr 270 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
185172, 176, 1843eqtr3d 2128 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
186185expr 367 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
187186exlimdv 1747 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
188187expimpd 355 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
189 fsumadd.1 . . 3 (𝜑𝐴 ∈ Fin)
190 fz1f1o 10728 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
191189, 190syl 14 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
19212, 188, 191mpjaod 673 1 (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  DECID wdc 780  w3a 924   = wceq 1289  wex 1426  wcel 1438  wral 2359  c0 3284  ifcif 3389   class class class wbr 3837  cmpt 3891  ccom 4432  wf 4998  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  Fincfn 6437  cc 7327  0cc0 7329  1c1 7330   + caddc 7332  cle 7502  cn 8394  cz 8720  cuz 8988  ...cfz 9393  seqcseq4 9816  chash 10148  Σcsu 10706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631  df-isum 10707
This theorem is referenced by:  fsumsplit  10764  fsumsub  10809  binomlem  10839
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