ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsumadd GIF version

Theorem fsumadd 11398
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 8088 . . . . 5 (0 + 0) = 0
2 sum0 11380 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
3 sum0 11380 . . . . . 6 Σ𝑘 ∈ ∅ 𝐶 = 0
42, 3oveq12i 5881 . . . . 5 𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶) = (0 + 0)
5 sum0 11380 . . . . 5 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = 0
61, 4, 53eqtr4ri 2209 . . . 4 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶)
7 sumeq1 11347 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = Σ𝑘 ∈ ∅ (𝐵 + 𝐶))
8 sumeq1 11347 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
9 sumeq1 11347 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 5887 . . . 4 (𝐴 = ∅ → (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2236 . . 3 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
1211a1i 9 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
13 simprl 529 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 9552 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14eleqtrdi 2270 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 eqid 2177 . . . . . . . . . 10 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
17 breq1 4003 . . . . . . . . . . 11 (𝑗 = 𝑛 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑛 ≤ (♯‘𝐴)))
18 fveq2 5511 . . . . . . . . . . 11 (𝑗 = 𝑛 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛))
1917, 18ifbieq1d 3556 . . . . . . . . . 10 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))
20 elnnuz 9553 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
2120biimpri 133 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ ℕ)
2221adantl 277 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → 𝑛 ∈ ℕ)
23 fsumadd.2 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2423adantlr 477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
2524fmpttd 5667 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
26 simprr 531 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
27 f1of 5457 . . . . . . . . . . . . . . 15 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2826, 27syl 14 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
29 fco 5377 . . . . . . . . . . . . . 14 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
3025, 28, 29syl2anc 411 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
3130ad2antrr 488 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
32 1zzd 9269 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
3313ad2antrr 488 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ)
3433nnzd 9363 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (♯‘𝐴) ∈ ℤ)
35 eluzelz 9526 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → 𝑛 ∈ ℤ)
3635ad2antlr 489 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ ℤ)
3732, 34, 363jca 1177 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ))
38 eluzle 9529 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘1) → 1 ≤ 𝑛)
3938ad2antlr 489 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 1 ≤ 𝑛)
40 simpr 110 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ≤ (♯‘𝐴))
4139, 40jca 306 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (1 ≤ 𝑛𝑛 ≤ (♯‘𝐴)))
42 elfz2 10002 . . . . . . . . . . . . 13 (𝑛 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (1 ≤ 𝑛𝑛 ≤ (♯‘𝐴))))
4337, 41, 42sylanbrc 417 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑛 ∈ (1...(♯‘𝐴)))
4431, 43ffvelcdmd 5648 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
45 0cnd 7941 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → 0 ∈ ℂ)
4622nnzd 9363 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → 𝑛 ∈ ℤ)
4713adantr 276 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℕ)
4847nnzd 9363 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (♯‘𝐴) ∈ ℤ)
49 zdcle 9318 . . . . . . . . . . . 12 ((𝑛 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑛 ≤ (♯‘𝐴))
5046, 48, 49syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → DECID 𝑛 ≤ (♯‘𝐴))
5144, 45, 50ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
5216, 19, 22, 51fvmptd3 5605 . . . . . . . . 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))
5352, 51eqeltrd 2254 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
54 eqid 2177 . . . . . . . . . 10 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
55 fveq2 5511 . . . . . . . . . . 11 (𝑗 = 𝑛 → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛))
5617, 55ifbieq1d 3556 . . . . . . . . . 10 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0))
57 fsumadd.3 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
5857adantlr 477 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
5958fmpttd 5667 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
6059ad2antrr 488 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
6128ad2antrr 488 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
62 fco 5377 . . . . . . . . . . . . 13 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6360, 61, 62syl2anc 411 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
6463, 43ffvelcdmd 5648 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
6564, 45, 50ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
6654, 56, 22, 65fvmptd3 5605 . . . . . . . . 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))
6766, 65eqeltrd 2254 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → ((𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))‘𝑛) ∈ ℂ)
68 simpll 527 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
6928ffvelcdmda 5647 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
70 simpr 110 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → 𝑘𝐴)
7123, 57addcld 7967 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
72 eqid 2177 . . . . . . . . . . . . . . . . . . 19 (𝑘𝐴 ↦ (𝐵 + 𝐶)) = (𝑘𝐴 ↦ (𝐵 + 𝐶))
7372fvmpt2 5595 . . . . . . . . . . . . . . . . . 18 ((𝑘𝐴 ∧ (𝐵 + 𝐶) ∈ ℂ) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
7470, 71, 73syl2anc 411 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
75 eqid 2177 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
7675fvmpt2 5595 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
7770, 23, 76syl2anc 411 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
78 eqid 2177 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
7978fvmpt2 5595 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
8070, 57, 79syl2anc 411 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
8177, 80oveq12d 5887 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 + 𝐶))
8274, 81eqtr4d 2213 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
8382ralrimiva 2550 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
8483ad2antrr 488 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
85 nffvmpt1 5522 . . . . . . . . . . . . . . . 16 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛))
86 nffvmpt1 5522 . . . . . . . . . . . . . . . . 17 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
87 nfcv 2319 . . . . . . . . . . . . . . . . 17 𝑘 +
88 nffvmpt1 5522 . . . . . . . . . . . . . . . . 17 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
8986, 87, 88nfov 5899 . . . . . . . . . . . . . . . 16 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9085, 89nfeq 2327 . . . . . . . . . . . . . . 15 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
91 fveq2 5511 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
92 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
93 fveq2 5511 . . . . . . . . . . . . . . . . 17 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
9492, 93oveq12d 5887 . . . . . . . . . . . . . . . 16 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
9591, 94eqeq12d 2192 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
9690, 95rspc 2835 . . . . . . . . . . . . . 14 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
9769, 84, 96sylc 62 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
98 fvco3 5583 . . . . . . . . . . . . . 14 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
9928, 98sylan 283 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
100 fvco3 5583 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
10128, 100sylan 283 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
102 fvco3 5583 . . . . . . . . . . . . . . 15 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
10328, 102sylan 283 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
104101, 103oveq12d 5887 . . . . . . . . . . . . 13 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
10597, 99, 1043eqtr4d 2220 . . . . . . . . . . . 12 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
10668, 43, 105syl2anc 411 . . . . . . . . . . 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)))
10740iftrued 3541 . . . . . . . . . . 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))
10840iftrued 3541 . . . . . . . . . . . 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))
10940iftrued 3541 . . . . . . . . . . . 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))
110108, 109oveq12d 5887 . . . . . . . . . . 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)))
111106, 107, 1103eqtr4d 2220 . . . . . . . . . 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)))
1121eqcomi 2181 . . . . . . . . . . 11 0 = (0 + 0)
113 simpr 110 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → ¬ 𝑛 ≤ (♯‘𝐴))
114113iffalsed 3544 . . . . . . . . . . 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)
115113iffalsed 3544 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = 0)
116113iffalsed 3544 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ ¬ 𝑛 ≤ (♯‘𝐴)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) = 0)
117115, 116oveq12d 5887 . . . . . . . . . . 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))
118112, 114, 1173eqtr4a 2236 . . . . . . . . . 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)))
119 exmiddc 836 . . . . . . . . . . 11 (DECID 𝑛 ≤ (♯‘𝐴) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
12050, 119syl 14 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → (𝑛 ≤ (♯‘𝐴) ∨ ¬ 𝑛 ≤ (♯‘𝐴)))
121111, 118, 120mpjaodan 798 . . . . . . . . 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)))
122 eqid 2177 . . . . . . . . . 10 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
123 fveq2 5511 . . . . . . . . . . 11 (𝑗 = 𝑛 → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗) = (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛))
12417, 123ifbieq1d 3556 . . . . . . . . . 10 (𝑗 = 𝑛 → if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0) = if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))
12571fmpttd 5667 . . . . . . . . . . . . . 14 (𝜑 → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
126125ad3antrrr 492 . . . . . . . . . . . . 13 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
127 fco 5377 . . . . . . . . . . . . 13 (((𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
128126, 61, 127syl2anc 411 . . . . . . . . . . . 12 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
129128, 43ffvelcdmd 5648 . . . . . . . . . . 11 ((((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) ∧ 𝑛 ≤ (♯‘𝐴)) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) ∈ ℂ)
130129, 45, 50ifcldadc 3563 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (ℤ‘1)) → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) ∈ ℂ)
131122, 124, 22, 130fvmptd3 5605 . . . . . . . . 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))
13252, 66oveq12d 5887 . . . . . . . . 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)))
133121, 131, 1323eqtr4d 2220 . . . . . . . 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)))
13415, 53, 67, 133ser3add 10491 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)) = ((seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))))
135 fveq2 5511 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
13624, 58addcld 7967 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
137136fmpttd 5667 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
138137ffvelcdmda 5647 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) ∈ ℂ)
139135, 13, 26, 138, 99fsum3 11379 . . . . . . . 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)))‘(♯‘𝐴)))
140 breq1 4003 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
141 fveq2 5511 . . . . . . . . . . . 12 (𝑛 = 𝑗 → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗))
142140, 141ifbieq1d 3556 . . . . . . . . . . 11 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
143142cbvmptv 4096 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))
144 seqeq3 10436 . . . . . . . . . 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))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0))))
145143, 144ax-mp 5 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))
146145fveq1i 5512 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
147139, 146eqtrdi 2226 . . . . . . 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)))‘(♯‘𝐴)))
148 fveq2 5511 . . . . . . . . . 10 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
14925ffvelcdmda 5647 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
150148, 13, 26, 149, 101fsum3 11379 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
151 fveq2 5511 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗))
152140, 151ifbieq1d 3556 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
153152cbvmptv 4096 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))
154 seqeq3 10436 . . . . . . . . . . 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))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0))))
155153, 154ax-mp 5 . . . . . . . . . 10 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)))
156155fveq1i 5512 . . . . . . . . 9 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
157150, 156eqtrdi 2226 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)))
158 fveq2 5511 . . . . . . . . . 10 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
15959ffvelcdmda 5647 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
160158, 13, 26, 159, 103fsum3 11379 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)))
161 fveq2 5511 . . . . . . . . . . . . 13 (𝑛 = 𝑗 → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗))
162140, 161ifbieq1d 3556 . . . . . . . . . . . 12 (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0) = if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
163162cbvmptv 4096 . . . . . . . . . . 11 (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))
164 seqeq3 10436 . . . . . . . . . . 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))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0))))
165163, 164ax-mp 5 . . . . . . . . . 10 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0))) = seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)))
166165fveq1i 5512 . . . . . . . . 9 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛), 0)))‘(♯‘𝐴)) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))
167160, 166eqtrdi 2226 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴)))
168157, 167oveq12d 5887 . . . . . . 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)))‘(♯‘𝐴)) + (seq1( + , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑗), 0)))‘(♯‘𝐴))))
169134, 147, 1683eqtr4d 2220 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
17071ralrimiva 2550 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 (𝐵 + 𝐶) ∈ ℂ)
171 sumfct 11366 . . . . . . . 8 (∀𝑘𝐴 (𝐵 + 𝐶) ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
172170, 171syl 14 . . . . . . 7 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
173172adantr 276 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶))
17423ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
175 sumfct 11366 . . . . . . . . 9 (∀𝑘𝐴 𝐵 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
176174, 175syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵)
17757ralrimiva 2550 . . . . . . . . 9 (𝜑 → ∀𝑘𝐴 𝐶 ∈ ℂ)
178 sumfct 11366 . . . . . . . . 9 (∀𝑘𝐴 𝐶 ∈ ℂ → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
179177, 178syl 14 . . . . . . . 8 (𝜑 → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶)
180176, 179oveq12d 5887 . . . . . . 7 (𝜑 → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
181180adantr 276 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
182169, 173, 1813eqtr3d 2218 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
183182expr 375 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
184183exlimdv 1819 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
185184expimpd 363 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
186 fsumadd.1 . . 3 (𝜑𝐴 ∈ Fin)
187 fz1f1o 11367 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
188186, 187syl 14 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
18912, 185, 188mpjaod 718 1 (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 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 3422  ifcif 3534   class class class wbr 4000  cmpt 4061  ccom 4627  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  cc 7800  0cc0 7802  1c1 7803   + caddc 7805  cle 7983  cn 8908  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  Σcsu 11345
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346
This theorem is referenced by:  fsumsplit  11399  fsumsub  11444  binomlem  11475  pcbc  12332
  Copyright terms: Public domain W3C validator