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Theorem fsumrelem 14746
 Description: Lemma for fsumre 14747, fsumim 14748, and fsumcj 14749. (Contributed by Mario Carneiro, 25-Jul-2014.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
fsumre.1 (𝜑𝐴 ∈ Fin)
fsumre.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
fsumrelem.3 𝐹:ℂ⟶ℂ
fsumrelem.4 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
Assertion
Ref Expression
fsumrelem (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝑥,𝐵,𝑦   𝑘,𝐹,𝑥,𝑦   𝜑,𝑘,𝑥,𝑦
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fsumrelem
Dummy variables 𝑓 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 10238 . . . . . . . 8 0 ∈ ℂ
2 fsumrelem.3 . . . . . . . . 9 𝐹:ℂ⟶ℂ
32ffvelrni 6503 . . . . . . . 8 (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
41, 3ax-mp 5 . . . . . . 7 (𝐹‘0) ∈ ℂ
54addid1i 10429 . . . . . 6 ((𝐹‘0) + 0) = (𝐹‘0)
6 fvoveq1 6819 . . . . . . . . 9 (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7 fveq2 6333 . . . . . . . . . 10 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
87oveq1d 6811 . . . . . . . . 9 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹𝑦)))
96, 8eqeq12d 2786 . . . . . . . 8 (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦))))
10 oveq2 6804 . . . . . . . . . . 11 (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11 00id 10417 . . . . . . . . . . 11 (0 + 0) = 0
1210, 11syl6eq 2821 . . . . . . . . . 10 (𝑦 = 0 → (0 + 𝑦) = 0)
1312fveq2d 6337 . . . . . . . . 9 (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14 fveq2 6333 . . . . . . . . . 10 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
1514oveq2d 6812 . . . . . . . . 9 (𝑦 = 0 → ((𝐹‘0) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹‘0)))
1613, 15eqeq12d 2786 . . . . . . . 8 (𝑦 = 0 → ((𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦)) ↔ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))))
17 fsumrelem.4 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
189, 16, 17vtocl2ga 3425 . . . . . . 7 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
191, 1, 18mp2an 672 . . . . . 6 (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
205, 19eqtr2i 2794 . . . . 5 ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
214, 4, 1addcani 10435 . . . . 5 (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
2220, 21mpbi 220 . . . 4 (𝐹‘0) = 0
23 sumeq1 14627 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24 sum0 14660 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
2523, 24syl6eq 2821 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
2625fveq2d 6337 . . . 4 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = (𝐹‘0))
27 sumeq1 14627 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = Σ𝑘 ∈ ∅ (𝐹𝐵))
28 sum0 14660 . . . . 5 Σ𝑘 ∈ ∅ (𝐹𝐵) = 0
2927, 28syl6eq 2821 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = 0)
3022, 26, 293eqtr4a 2831 . . 3 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
3130a1i 11 . 2 (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
32 addcl 10224 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3332adantl 467 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
34 fsumre.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3534fmpttd 6530 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
3635adantr 466 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
37 simprr 756 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
38 f1of 6279 . . . . . . . . . . 11 (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴𝑓:(1...(♯‘𝐴))⟶𝐴)
3937, 38syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
40 fco 6199 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
4136, 39, 40syl2anc 573 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
4241ffvelrnda 6504 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43 simprl 754 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
44 nnuz 11930 . . . . . . . . 9 ℕ = (ℤ‘1)
4543, 44syl6eleq 2860 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
4617adantl 467 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
4739ffvelrnda 6504 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
48 simpr 471 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
49 eqid 2771 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5049fvmpt2 6435 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5148, 34, 50syl2anc 573 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5251fveq2d 6337 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹𝐵))
53 fvex 6344 . . . . . . . . . . . . . 14 (𝐹𝐵) ∈ V
54 eqid 2771 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐹𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵))
5554fvmpt2 6435 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐹𝐵) ∈ V) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5648, 53, 55sylancl 574 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5752, 56eqtr4d 2808 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5857ralrimiva 3115 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5958ad2antrr 705 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
60 nfcv 2913 . . . . . . . . . . . . 13 𝑘𝐹
61 nffvmpt1 6342 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑥))
6260, 61nffv 6341 . . . . . . . . . . . 12 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥)))
63 nffvmpt1 6342 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
6462, 63nfeq 2925 . . . . . . . . . . 11 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
65 fveq2 6333 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
6665fveq2d 6337 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
67 fveq2 6333 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
6866, 67eqeq12d 2786 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → ((𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) ↔ (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
6964, 68rspc 3454 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
7047, 59, 69sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
71 fvco3 6419 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7239, 71sylan 569 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7372fveq2d 6337 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
74 fvco3 6419 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7539, 74sylan 569 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7670, 73, 753eqtr4d 2815 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥))
7733, 42, 45, 46, 76seqhomo 13055 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
78 fveq2 6333 . . . . . . . . 9 (𝑚 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7936ffvelrnda 6504 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8078, 43, 37, 79, 72fsum 14659 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
8180fveq2d 6337 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
82 fveq2 6333 . . . . . . . 8 (𝑚 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
832ffvelrni 6503 . . . . . . . . . . . 12 (𝐵 ∈ ℂ → (𝐹𝐵) ∈ ℂ)
8434, 83syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐹𝐵) ∈ ℂ)
8584fmpttd 6530 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8685adantr 466 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8786ffvelrnda 6504 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) ∈ ℂ)
8882, 43, 37, 87, 75fsum 14659 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
8977, 81, 883eqtr4d 2815 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚))
90 sumfc 14648 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
9190fveq2i 6336 . . . . . 6 (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘Σ𝑘𝐴 𝐵)
92 sumfc 14648 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = Σ𝑘𝐴 (𝐹𝐵)
9389, 91, 923eqtr3g 2828 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
9493expr 444 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9594exlimdv 2013 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9695expimpd 441 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
97 fsumre.1 . . 3 (𝜑𝐴 ∈ Fin)
98 fz1f1o 14649 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9997, 98syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
10031, 96, 99mpjaod 849 1 (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 836   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  Vcvv 3351  ∅c0 4063   ↦ cmpt 4864   ∘ ccom 5254  ⟶wf 6026  –1-1-onto→wf1o 6029  ‘cfv 6030  (class class class)co 6796  Fincfn 8113  ℂcc 10140  0cc0 10142  1c1 10143   + caddc 10145  ℕcn 11226  ℤ≥cuz 11893  ...cfz 12533  seqcseq 13008  ♯chash 13321  Σcsu 14624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-oi 8575  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-sum 14625 This theorem is referenced by:  fsumre  14747  fsumim  14748  fsumcj  14749
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