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Theorem fsumadd 14669
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 10403 . . . . 5 (0 + 0) = 0
2 sum0 14651 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
3 sum0 14651 . . . . . 6 Σ𝑘 ∈ ∅ 𝐶 = 0
42, 3oveq12i 6825 . . . . 5 𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶) = (0 + 0)
5 sum0 14651 . . . . 5 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = 0
61, 4, 53eqtr4ri 2793 . . . 4 Σ𝑘 ∈ ∅ (𝐵 + 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶)
7 sumeq1 14618 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = Σ𝑘 ∈ ∅ (𝐵 + 𝐶))
8 sumeq1 14618 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
9 sumeq1 14618 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
108, 9oveq12d 6831 . . . 4 (𝐴 = ∅ → (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶) = (Σ𝑘 ∈ ∅ 𝐵 + Σ𝑘 ∈ ∅ 𝐶))
116, 7, 103eqtr4a 2820 . . 3 (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
1211a1i 11 . 2 (𝜑 → (𝐴 = ∅ → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
13 simprl 811 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
14 nnuz 11916 . . . . . . . . 9 ℕ = (ℤ‘1)
1513, 14syl6eleq 2849 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
16 fsumadd.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1716adantlr 753 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
18 eqid 2760 . . . . . . . . . . 11 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
1917, 18fmptd 6548 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
20 simprr 813 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
21 f1of 6298 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
2220, 21syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
23 fco 6219 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2419, 22, 23syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
2524ffvelrnda 6522 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) ∈ ℂ)
26 fsumadd.3 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)
2726adantlr 753 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐶 ∈ ℂ)
28 eqid 2760 . . . . . . . . . . 11 (𝑘𝐴𝐶) = (𝑘𝐴𝐶)
2927, 28fmptd 6548 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐶):𝐴⟶ℂ)
30 fco 6219 . . . . . . . . . 10 (((𝑘𝐴𝐶):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
3129, 22, 30syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐶) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
3231ffvelrnda 6522 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) ∈ ℂ)
3322ffvelrnda 6522 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑓𝑛) ∈ 𝐴)
34 ovex 6841 . . . . . . . . . . . . . . 15 (𝐵 + 𝐶) ∈ V
35 eqid 2760 . . . . . . . . . . . . . . . 16 (𝑘𝐴 ↦ (𝐵 + 𝐶)) = (𝑘𝐴 ↦ (𝐵 + 𝐶))
3635fvmpt2 6453 . . . . . . . . . . . . . . 15 ((𝑘𝐴 ∧ (𝐵 + 𝐶) ∈ V) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
3734, 36mpan2 709 . . . . . . . . . . . . . 14 (𝑘𝐴 → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
3837adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (𝐵 + 𝐶))
39 simpr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
4018fvmpt2 6453 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4139, 16, 40syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
4228fvmpt2 6453 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐶 ∈ ℂ) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4339, 26, 42syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐶)‘𝑘) = 𝐶)
4441, 43oveq12d 6831 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (𝐵 + 𝐶))
4538, 44eqtr4d 2797 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
4645ralrimiva 3104 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
4746ad2antrr 764 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)))
48 nffvmpt1 6360 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛))
49 nffvmpt1 6360 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑛))
50 nfcv 2902 . . . . . . . . . . . . 13 𝑘 +
51 nffvmpt1 6360 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐶)‘(𝑓𝑛))
5249, 50, 51nfov 6839 . . . . . . . . . . . 12 𝑘(((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5348, 52nfeq 2914 . . . . . . . . . . 11 𝑘((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))
54 fveq2 6352 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
55 fveq2 6352 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
56 fveq2 6352 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑘) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
5755, 56oveq12d 6831 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑛) → (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛))))
5854, 57eqeq12d 2775 . . . . . . . . . . 11 (𝑘 = (𝑓𝑛) → (((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) ↔ ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
5953, 58rspc 3443 . . . . . . . . . 10 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑘) = (((𝑘𝐴𝐵)‘𝑘) + ((𝑘𝐴𝐶)‘𝑘)) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)) = (((𝑘𝐴𝐵)‘(𝑓𝑛)) + ((𝑘𝐴𝐶)‘(𝑓𝑛)))))
6033, 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 6437 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
6222, 61sylan 489 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓)‘𝑛) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
63 fvco3 6437 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
6422, 63sylan 489 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
65 fvco3 6437 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6622, 65sylan 489 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐶) ∘ 𝑓)‘𝑛) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
6764, 66oveq12d 6831 . . . . . . . . 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 2804 . . . . . . . 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, 25, 32, 68seradd 13037 . . . . . . 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 6352 . . . . . . . 8 (𝑚 = (𝑓𝑛) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘(𝑓𝑛)))
7117, 27addcld 10251 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑘𝐴) → (𝐵 + 𝐶) ∈ ℂ)
7271, 35fmptd 6548 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐵 + 𝐶)):𝐴⟶ℂ)
7372ffvelrnda 6522 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) ∈ ℂ)
7470, 13, 20, 73, 62fsum 14650 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐵 + 𝐶)) ∘ 𝑓))‘(♯‘𝐴)))
75 fveq2 6352 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑛)))
7619ffvelrnda 6522 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7775, 13, 20, 76, 64fsum 14650 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
78 fveq2 6352 . . . . . . . . 9 (𝑚 = (𝑓𝑛) → ((𝑘𝐴𝐶)‘𝑚) = ((𝑘𝐴𝐶)‘(𝑓𝑛)))
7929ffvelrnda 6522 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐶)‘𝑚) ∈ ℂ)
8078, 13, 20, 79, 66fsum 14650 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = (seq1( + , ((𝑘𝐴𝐶) ∘ 𝑓))‘(♯‘𝐴)))
8177, 80oveq12d 6831 . . . . . . 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 2804 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = (Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)))
83 sumfc 14639 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐵 + 𝐶))‘𝑚) = Σ𝑘𝐴 (𝐵 + 𝐶)
84 sumfc 14639 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
85 sumfc 14639 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚) = Σ𝑘𝐴 𝐶
8684, 85oveq12i 6825 . . . . . 6 𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) + Σ𝑚𝐴 ((𝑘𝐴𝐶)‘𝑚)) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)
8782, 83, 863eqtr3g 2817 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
8887expr 644 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
8988exlimdv 2010 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶)))
9089expimpd 630 . 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 14640 . . 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 395 1 (𝜑 → Σ𝑘𝐴 (𝐵 + 𝐶) = (Σ𝑘𝐴 𝐵 + Σ𝑘𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1632  wex 1853  wcel 2139  wral 3050  Vcvv 3340  c0 4058  cmpt 4881  ccom 5270  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  Fincfn 8121  cc 10126  0cc0 10128  1c1 10129   + caddc 10131  cn 11212  cuz 11879  ...cfz 12519  seqcseq 12995  chash 13311  Σcsu 14615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-sum 14616
This theorem is referenced by:  fsumsplit  14670  fsumsub  14719  binomlem  14760  binomfallfaclem2  14970  pwp1fsum  15316  pcbc  15806  csbren  23382  trirn  23383  ovollb2lem  23456  ovoliunlem1  23470  itg1addlem5  23666  itgsplit  23801  plyaddlem1  24168  basellem8  25013  logfaclbnd  25146  dchrvmasum2if  25385  mudivsum  25418  logsqvma  25430  selberglem1  25433  selberglem2  25434  selberg  25436  selberg2  25439  selberg3lem1  25445  selberg4  25449  pntsval2  25464  ax5seglem9  26016  finsumvtxdg2ssteplem4  26654  dvnmul  40661  dirkertrigeqlem2  40819  sge0xaddlem1  41153  sge0xaddlem2  41154  hoidmvlelem2  41316  altgsumbcALT  42641
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