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Theorem fsumrelem 14461
Description: Lemma for fsumre 14462, fsumim 14463, and fsumcj 14464. (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 9977 . . . . . . . 8 0 ∈ ℂ
2 fsumrelem.3 . . . . . . . . 9 𝐹:ℂ⟶ℂ
32ffvelrni 6315 . . . . . . . 8 (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
41, 3ax-mp 5 . . . . . . 7 (𝐹‘0) ∈ ℂ
54addid1i 10168 . . . . . 6 ((𝐹‘0) + 0) = (𝐹‘0)
6 oveq1 6612 . . . . . . . . . 10 (𝑥 = 0 → (𝑥 + 𝑦) = (0 + 𝑦))
76fveq2d 6154 . . . . . . . . 9 (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
8 fveq2 6150 . . . . . . . . . 10 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
98oveq1d 6620 . . . . . . . . 9 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹𝑦)))
107, 9eqeq12d 2641 . . . . . . . 8 (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦))))
11 oveq2 6613 . . . . . . . . . . 11 (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
12 00id 10156 . . . . . . . . . . 11 (0 + 0) = 0
1311, 12syl6eq 2676 . . . . . . . . . 10 (𝑦 = 0 → (0 + 𝑦) = 0)
1413fveq2d 6154 . . . . . . . . 9 (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
15 fveq2 6150 . . . . . . . . . 10 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
1615oveq2d 6621 . . . . . . . . 9 (𝑦 = 0 → ((𝐹‘0) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹‘0)))
1714, 16eqeq12d 2641 . . . . . . . 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 3265 . . . . . . 7 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
201, 1, 19mp2an 707 . . . . . 6 (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
215, 20eqtr2i 2649 . . . . 5 ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
224, 4, 1addcani 10174 . . . . 5 (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
2321, 22mpbi 220 . . . 4 (𝐹‘0) = 0
24 sumeq1 14348 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
25 sum0 14380 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
2624, 25syl6eq 2676 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
2726fveq2d 6154 . . . 4 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = (𝐹‘0))
28 sumeq1 14348 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = Σ𝑘 ∈ ∅ (𝐹𝐵))
29 sum0 14380 . . . . 5 Σ𝑘 ∈ ∅ (𝐹𝐵) = 0
3028, 29syl6eq 2676 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = 0)
3123, 27, 303eqtr4a 2686 . . 3 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
3231a1i 11 . 2 (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
33 addcl 9963 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3433adantl 482 . . . . . . . 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 2626 . . . . . . . . . . . 12 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
3735, 36fmptd 6341 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
3837adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
39 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)
40 f1of 6096 . . . . . . . . . . 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 6017 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(#‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
4338, 41, 42syl2anc 692 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(#‘𝐴))⟶ℂ)
4443ffvelrnda 6316 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
45 simprl 793 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ ℕ)
46 nnuz 11667 . . . . . . . . 9 ℕ = (ℤ‘1)
4745, 46syl6eleq 2714 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (#‘𝐴) ∈ (ℤ‘1))
4818adantl 482 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
4941ffvelrnda 6316 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
50 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
5136fvmpt2 6249 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5250, 35, 51syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5352fveq2d 6154 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹𝐵))
54 fvex 6160 . . . . . . . . . . . . . 14 (𝐹𝐵) ∈ V
55 eqid 2626 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐹𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵))
5655fvmpt2 6249 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐹𝐵) ∈ V) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5750, 54, 56sylancl 693 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5853, 57eqtr4d 2663 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5958ralrimiva 2965 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
6059ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
61 nfcv 2767 . . . . . . . . . . . . 13 𝑘𝐹
62 nffvmpt1 6158 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑥))
6361, 62nffv 6157 . . . . . . . . . . . 12 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥)))
64 nffvmpt1 6158 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
6563, 64nfeq 2778 . . . . . . . . . . 11 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
66 fveq2 6150 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑘) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
6766fveq2d 6154 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
68 fveq2 6150 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
6967, 68eqeq12d 2641 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → ((𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) ↔ (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
7065, 69rspc 3294 . . . . . . . . . 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 6233 . . . . . . . . . . 11 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7341, 72sylan 488 . . . . . . . . . 10 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7473fveq2d 6154 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
75 fvco3 6233 . . . . . . . . . 10 ((𝑓:(1...(#‘𝐴))⟶𝐴𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7641, 75sylan 488 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(#‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7771, 74, 763eqtr4d 2670 . . . . . . . 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 12785 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(#‘𝐴)))
79 fveq2 6150 . . . . . . . . 9 (𝑚 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
8038ffvelrnda 6316 . . . . . . . . 9 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
8179, 45, 39, 80, 73fsum 14379 . . . . . . . 8 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴)))
8281fveq2d 6154 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(#‘𝐴))))
83 fveq2 6150 . . . . . . . 8 (𝑚 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
842ffvelrni 6315 . . . . . . . . . . . 12 (𝐵 ∈ ℂ → (𝐹𝐵) ∈ ℂ)
8535, 84syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐹𝐵) ∈ ℂ)
8685, 55fmptd 6341 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8786adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8887ffvelrnda 6316 . . . . . . . 8 (((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) ∈ ℂ)
8983, 45, 39, 88, 76fsum 14379 . . . . . . 7 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(#‘𝐴)))
9078, 82, 893eqtr4d 2670 . . . . . 6 ((𝜑 ∧ ((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚))
91 sumfc 14368 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
9291fveq2i 6153 . . . . . 6 (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘Σ𝑘𝐴 𝐵)
93 sumfc 14368 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = Σ𝑘𝐴 (𝐹𝐵)
9490, 92, 933eqtr3g 2683 . . . . 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 1863 . . 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 14369 . . 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 396 1 (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1992  wral 2912  Vcvv 3191  c0 3896  cmpt 4678  ccom 5083  wf 5846  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Fincfn 7900  cc 9879  0cc0 9881  1c1 9882   + caddc 9884  cn 10965  cuz 11631  ...cfz 12265  seqcseq 12738  #chash 13054  Σcsu 14345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346
This theorem is referenced by:  fsumre  14462  fsumim  14463  fsumcj  14464
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