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Theorem fsumrelem 14583
Description: Lemma for fsumre 14584, fsumim 14585, and fsumcj 14586. (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 10070 . . . . . . . 8 0 ∈ ℂ
2 fsumrelem.3 . . . . . . . . 9 𝐹:ℂ⟶ℂ
32ffvelrni 6398 . . . . . . . 8 (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
41, 3ax-mp 5 . . . . . . 7 (𝐹‘0) ∈ ℂ
54addid1i 10261 . . . . . 6 ((𝐹‘0) + 0) = (𝐹‘0)
6 oveq1 6697 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 + 𝑦) = (0 + 𝑦))
76fveq2d 6233 . . . . . . . . 9 (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
8 fveq2 6229 . . . . . . . . . 10 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
98oveq1d 6705 . . . . . . . . 9 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹𝑦)))
107, 9eqeq12d 2666 . . . . . . . 8 (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦))))
11 oveq2 6698 . . . . . . . . . . 11 (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
12 00id 10249 . . . . . . . . . . 11 (0 + 0) = 0
1311, 12syl6eq 2701 . . . . . . . . . 10 (𝑦 = 0 → (0 + 𝑦) = 0)
1413fveq2d 6233 . . . . . . . . 9 (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
15 fveq2 6229 . . . . . . . . . 10 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
1615oveq2d 6706 . . . . . . . . 9 (𝑦 = 0 → ((𝐹‘0) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹‘0)))
1714, 16eqeq12d 2666 . . . . . . . 8 (𝑦 = 0 → ((𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦)) ↔ (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))))
18 fsumrelem.4 . . . . . . . 8 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
1910, 17, 18vtocl2ga 3305 . . . . . . 7 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
201, 1, 19mp2an 708 . . . . . 6 (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
215, 20eqtr2i 2674 . . . . 5 ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
224, 4, 1addcani 10267 . . . . 5 (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
2321, 22mpbi 220 . . . 4 (𝐹‘0) = 0
24 sumeq1 14463 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
25 sum0 14496 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
2624, 25syl6eq 2701 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
2726fveq2d 6233 . . . 4 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = (𝐹‘0))
28 sumeq1 14463 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = Σ𝑘 ∈ ∅ (𝐹𝐵))
29 sum0 14496 . . . . 5 Σ𝑘 ∈ ∅ (𝐹𝐵) = 0
3028, 29syl6eq 2701 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = 0)
3123, 27, 303eqtr4a 2711 . . 3 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
3231a1i 11 . 2 (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
33 addcl 10056 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3433adantl 481 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 + 𝑦) ∈ ℂ)
35 fsumre.2 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
36 eqid 2651 . . . . . . . . . . . 12 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
3735, 36fmptd 6425 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
3837adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
39 simprr 811 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)
40 f1of 6175 . . . . . . . . . . 11 (𝑓:(1...(#‘𝐴))–1-1-onto𝐴𝑓:(1...(#‘𝐴))⟶𝐴)
4139, 40syl 17 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))⟶𝐴)
42 fco 6096 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(#‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
4338, 41, 42syl2anc 694 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
4443ffvelrnda 6399 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
45 simprl 809 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
46 nnuz 11761 . . . . . . . . 9 ℕ = (ℤ‘1)
4745, 46syl6eleq 2740 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
4818adantl 481 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
4941ffvelrnda 6399 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
50 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
5136fvmpt2 6330 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5250, 35, 51syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5352fveq2d 6233 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹𝐵))
54 fvex 6239 . . . . . . . . . . . . . 14 (𝐹𝐵) ∈ V
55 eqid 2651 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐹𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵))
5655fvmpt2 6330 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐹𝐵) ∈ V) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5750, 54, 56sylancl 695 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5853, 57eqtr4d 2688 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5958ralrimiva 2995 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
6059ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
61 nfcv 2793 . . . . . . . . . . . . 13 𝑘𝐹
62 nffvmpt1 6237 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑥))
6361, 62nffv 6236 . . . . . . . . . . . 12 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥)))
64 nffvmpt1 6237 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
6563, 64nfeq 2805 . . . . . . . . . . 11 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
66 fveq2 6229 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
6766fveq2d 6233 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
68 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
6967, 68eqeq12d 2666 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → ((𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) ↔ (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
7065, 69rspc 3334 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
7149, 60, 70sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
72 fvco3 6314 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7341, 72sylan 487 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7473fveq2d 6233 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
75 fvco3 6314 . . . . . . . . . 10 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7641, 75sylan 487 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7771, 74, 763eqtr4d 2695 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥))
7834, 44, 47, 48, 77seqhomo 12888 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(#‘𝐴)))
79 fveq2 6229 . . . . . . . . 9 (𝑚 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
8038ffvelrnda 6399 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8179, 45, 39, 80, 73fsum 14495 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴)))
8281fveq2d 6233 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))))
83 fveq2 6229 . . . . . . . 8 (𝑚 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
842ffvelrni 6398 . . . . . . . . . . . 12 (𝐵 ∈ ℂ → (𝐹𝐵) ∈ ℂ)
8535, 84syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐹𝐵) ∈ ℂ)
8685, 55fmptd 6425 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8786adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8887ffvelrnda 6399 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) ∈ ℂ)
8983, 45, 39, 88, 76fsum 14495 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(#‘𝐴)))
9078, 82, 893eqtr4d 2695 . . . . . 6 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚))
91 sumfc 14484 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
9291fveq2i 6232 . . . . . 6 (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘Σ𝑘𝐴 𝐵)
93 sumfc 14484 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = Σ𝑘𝐴 (𝐹𝐵)
9490, 92, 933eqtr3g 2708 . . . . 5 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
9594expr 642 . . . 4 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9695exlimdv 1901 . . 3 ((𝜑 ∧ (#‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9796expimpd 628 . 2 (𝜑 → (((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
98 fsumre.1 . . 3 (𝜑𝐴 ∈ Fin)
99 fz1f1o 14485 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
10098, 99syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)))
10132, 97, 100mpjaod 395 1 (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231  c0 3948  cmpt 4762  ccom 5147  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  0cc0 9974  1c1 9975   + caddc 9977  cn 11058  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461
This theorem is referenced by:  fsumre  14584  fsumim  14585  fsumcj  14586
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