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Theorem fsumrelem 15692
Description: Lemma for fsumre 15693, fsumim 15694, and fsumcj 15695. (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 11147 . . . . . . . 8 0 ∈ ℂ
2 fsumrelem.3 . . . . . . . . 9 𝐹:ℂ⟶ℂ
32ffvelcdmi 7034 . . . . . . . 8 (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
41, 3ax-mp 5 . . . . . . 7 (𝐹‘0) ∈ ℂ
54addid1i 11342 . . . . . 6 ((𝐹‘0) + 0) = (𝐹‘0)
6 fvoveq1 7380 . . . . . . . . 9 (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7 fveq2 6842 . . . . . . . . . 10 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
87oveq1d 7372 . . . . . . . . 9 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹𝑦)))
96, 8eqeq12d 2752 . . . . . . . 8 (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦))))
10 oveq2 7365 . . . . . . . . . . 11 (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11 00id 11330 . . . . . . . . . . 11 (0 + 0) = 0
1210, 11eqtrdi 2792 . . . . . . . . . 10 (𝑦 = 0 → (0 + 𝑦) = 0)
1312fveq2d 6846 . . . . . . . . 9 (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14 fveq2 6842 . . . . . . . . . 10 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
1514oveq2d 7373 . . . . . . . . 9 (𝑦 = 0 → ((𝐹‘0) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹‘0)))
1613, 15eqeq12d 2752 . . . . . . . 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 3535 . . . . . . 7 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
191, 1, 18mp2an 690 . . . . . 6 (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
205, 19eqtr2i 2765 . . . . 5 ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
214, 4, 1addcani 11348 . . . . 5 (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
2220, 21mpbi 229 . . . 4 (𝐹‘0) = 0
23 sumeq1 15573 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24 sum0 15606 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
2523, 24eqtrdi 2792 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
2625fveq2d 6846 . . . 4 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = (𝐹‘0))
27 sumeq1 15573 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = Σ𝑘 ∈ ∅ (𝐹𝐵))
28 sum0 15606 . . . . 5 Σ𝑘 ∈ ∅ (𝐹𝐵) = 0
2927, 28eqtrdi 2792 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = 0)
3022, 26, 293eqtr4a 2802 . . 3 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
3130a1i 11 . 2 (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
32 addcl 11133 . . . . . . . . 9 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ)
3332adantl 482 . . . . . . . 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 7063 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
3635adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
37 simprr 771 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
38 f1of 6784 . . . . . . . . . . 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 6692 . . . . . . . . . 10 (((𝑘𝐴𝐵):𝐴⟶ℂ ∧ 𝑓:(1...(♯‘𝐴))⟶𝐴) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
4136, 39, 40syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ((𝑘𝐴𝐵) ∘ 𝑓):(1...(♯‘𝐴))⟶ℂ)
4241ffvelcdmda 7035 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43 simprl 769 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
44 nnuz 12806 . . . . . . . . 9 ℕ = (ℤ‘1)
4543, 44eleqtrdi 2848 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ (ℤ‘1))
4617adantl 482 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
4739ffvelcdmda 7035 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
48 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
49 eqid 2736 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5049fvmpt2 6959 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5148, 34, 50syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5251fveq2d 6846 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹𝐵))
53 fvex 6855 . . . . . . . . . . . . . 14 (𝐹𝐵) ∈ V
54 eqid 2736 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐹𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵))
5554fvmpt2 6959 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐹𝐵) ∈ V) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5648, 53, 55sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5752, 56eqtr4d 2779 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5857ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5958ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
60 nfcv 2907 . . . . . . . . . . . . 13 𝑘𝐹
61 nffvmpt1 6853 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑥))
6260, 61nffv 6852 . . . . . . . . . . . 12 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥)))
63 nffvmpt1 6853 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
6462, 63nfeq 2920 . . . . . . . . . . 11 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
65 2fveq3 6847 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
66 fveq2 6842 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
6765, 66eqeq12d 2752 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → ((𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) ↔ (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
6864, 67rspc 3569 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
6947, 59, 68sylc 65 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
70 fvco3 6940 . . . . . . . . . . 11 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7139, 70sylan 580 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7271fveq2d 6846 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
73 fvco3 6940 . . . . . . . . . 10 ((𝑓:(1...(♯‘𝐴))⟶𝐴𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7439, 73sylan 580 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
7569, 72, 743eqtr4d 2786 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓)‘𝑥))
7633, 42, 45, 46, 75seqhomo 13955 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
77 fveq2 6842 . . . . . . . . 9 (𝑚 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7836ffvelcdmda 7035 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7977, 43, 37, 78, 71fsum 15605 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
8079fveq2d 6846 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
81 fveq2 6842 . . . . . . . 8 (𝑚 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
822ffvelcdmi 7034 . . . . . . . . . . . 12 (𝐵 ∈ ℂ → (𝐹𝐵) ∈ ℂ)
8334, 82syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐹𝐵) ∈ ℂ)
8483fmpttd 7063 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8584adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8685ffvelcdmda 7035 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) ∈ ℂ)
8781, 43, 37, 86, 74fsum 15605 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
8876, 80, 873eqtr4d 2786 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚))
89 sumfc 15594 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
9089fveq2i 6845 . . . . . 6 (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘Σ𝑘𝐴 𝐵)
91 sumfc 15594 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = Σ𝑘𝐴 (𝐹𝐵)
9288, 90, 913eqtr3g 2799 . . . . 5 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
9392expr 457 . . . 4 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9493exlimdv 1936 . . 3 ((𝜑 ∧ (♯‘𝐴) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
9594expimpd 454 . 2 (𝜑 → (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
96 fsumre.1 . . 3 (𝜑𝐴 ∈ Fin)
97 fz1f1o 15595 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9896, 97syl 17 . 2 (𝜑 → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
9931, 95, 98mpjaod 858 1 (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wral 3064  Vcvv 3445  c0 4282  cmpt 5188  ccom 5637  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  cn 12153  cuz 12763  ...cfz 13424  seqcseq 13906  chash 14230  Σcsu 15570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571
This theorem is referenced by:  fsumre  15693  fsumim  15694  fsumcj  15695
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