MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumadd Structured version   Visualization version   GIF version

Theorem fsumadd 14757
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 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 00id 10465 . . . . 5 (0 + 0) = 0
2 sum0 14739 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
3 sum0 14739 . . . . . 6 Σ𝑘 ∈ ∅ 𝐶 = 0
42, 3oveq12i 6854 . . . . 5 𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶) = (0 + 0)
5 sum0 14739 . . . . 5 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = 0
61, 4, 53eqtr4ri 2798 . . . 4 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶)
7 sumeq1 14706 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = Σ𝑘 ∈ ∅ (𝐵 + 𝐶))
8 sumeq1 14706 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
9 sumeq1 14706 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 6860 . . . 4 (𝐴 = ∅ → (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2825 . . 3 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
1211a1i 11 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
13 simprl 787 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 11923 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14syl6eleq 2854 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 fsumadd.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1716adantlr 706 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
1817fmpttd 6575 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
19 simprr 789 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
20 f1of 6320 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2119, 20syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
22 fco 6240 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2318, 21, 22syl2anc 579 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2423ffvelrnda 6549 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
25 fsumadd.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2625adantlr 706 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
2726fmpttd 6575 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
28 fco 6240 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2927, 21, 28syl2anc 579 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
3029ffvelrnda 6549 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
3121ffvelrnda 6549 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
32 ovex 6874 . . . . . . . . . . . . . . 15 (𝐵 + 𝐶) ∈ V
33 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑘𝐴 ↦ (𝐵 + 𝐶)) = (𝑘𝐴 ↦ (𝐵 + 𝐶))
3433fvmpt2 6480 . . . . . . . . . . . . . . 15 ((𝑘𝐴 ∧ (𝐵 + 𝐶) ∈ V) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
3532, 34mpan2 682 . . . . . . . . . . . . . 14 (𝑘𝐴 → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
3635adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
37 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
38 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
3938fvmpt2 6480 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4037, 16, 39syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
41 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
4241fvmpt2 6480 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4337, 25, 42syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4440, 43oveq12d 6860 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 + 𝐶))
4536, 44eqtr4d 2802 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
4645ralrimiva 3113 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
4746ad2antrr 717 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
48 nffvmpt1 6386 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛))
49 nffvmpt1 6386 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
50 nfcv 2907 . . . . . . . . . . . . 13 𝑘 +
51 nffvmpt1 6386 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
5249, 50, 51nfov 6872 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5348, 52nfeq 2919 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
54 fveq2 6375 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
55 fveq2 6375 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
56 fveq2 6375 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5755, 56oveq12d 6860 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
5854, 57eqeq12d 2780 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
5953, 58rspc 3455 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
6031, 47, 59sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
61 fvco3 6464 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
6221, 61sylan 575 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
63 fvco3 6464 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6421, 63sylan 575 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
65 fvco3 6464 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6621, 65sylan 575 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6764, 66oveq12d 6860 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
6860, 62, 673eqtr4d 2809 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) + (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛)))
6915, 24, 30, 68seradd 13050 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (seq1( + , ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓))‘(♯‘𝐴)) = ((seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)) + (seq1( + , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴))))
70 fveq2 6375 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
7117, 26addcld 10313 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
7271fmpttd 6575 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
7372ffvelrnda 6549 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) ∈ ℂ)
7470, 13, 19, 73, 62fsum 14738 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
75 fveq2 6375 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7618ffvelrnda 6549 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7775, 13, 19, 76, 64fsum 14738 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
78 fveq2 6375 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
7927ffvelrnda 6549 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
8078, 13, 19, 79, 66fsum 14738 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴)))
8177, 80oveq12d 6860 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = ((seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)) + (seq1( + , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴))))
8269, 74, 813eqtr4d 2809 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
83 sumfc 14727 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶)
84 sumfc 14727 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
85 sumfc 14727 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶
8684, 85oveq12i 6854 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)
8782, 83, 863eqtr3g 2822 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
8887expr 448 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
8988exlimdv 2028 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
9089expimpd 445 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
91 fsumadd.1 . . 3 (𝜑𝐴 ∈ Fin)
92 fz1f1o 14728 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9391, 92syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9412, 90, 93mpjaod 886 1 (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 873   = wceq 1652  wex 1874  wcel 2155  wral 3055  Vcvv 3350  c0 4079  cmpt 4888  ccom 5281  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  0cc0 10189  1c1 10190   + caddc 10192  cn 11274  cuz 11886  ...cfz 12533  seqcseq 13008  chash 13321  Σcsu 14703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-clim 14506  df-sum 14704
This theorem is referenced by:  fsumsplit  14758  fsumsub  14806  binomlem  14847  binomfallfaclem2  15055  pwp1fsum  15398  pcbc  15885  csbren  23471  trirn  23472  ovollb2lem  23546  ovoliunlem1  23560  itg1addlem5  23758  itgsplit  23893  plyaddlem1  24260  basellem8  25105  logfaclbnd  25238  dchrvmasum2if  25477  mudivsum  25510  logsqvma  25522  selberglem1  25525  selberglem2  25526  selberg  25528  selberg2  25531  selberg3lem1  25537  selberg4  25541  pntsval2  25556  ax5seglem9  26108  finsumvtxdg2ssteplem4  26735  dvnmul  40728  dirkertrigeqlem2  40885  sge0xaddlem1  41219  sge0xaddlem2  41220  hoidmvlelem2  41382  altgsumbcALT  42732
  Copyright terms: Public domain W3C validator