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Theorem fsumrelem 15150
Description: Lemma for fsumre 15151, fsumim 15152, and fsumcj 15153. (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 10621 . . . . . . . 8 0 ∈ ℂ
2 fsumrelem.3 . . . . . . . . 9 𝐹:ℂ⟶ℂ
32ffvelrni 6842 . . . . . . . 8 (0 ∈ ℂ → (𝐹‘0) ∈ ℂ)
41, 3ax-mp 5 . . . . . . 7 (𝐹‘0) ∈ ℂ
54addid1i 10815 . . . . . 6 ((𝐹‘0) + 0) = (𝐹‘0)
6 fvoveq1 7168 . . . . . . . . 9 (𝑥 = 0 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(0 + 𝑦)))
7 fveq2 6663 . . . . . . . . . 10 (𝑥 = 0 → (𝐹𝑥) = (𝐹‘0))
87oveq1d 7160 . . . . . . . . 9 (𝑥 = 0 → ((𝐹𝑥) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹𝑦)))
96, 8eqeq12d 2834 . . . . . . . 8 (𝑥 = 0 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) + (𝐹𝑦)) ↔ (𝐹‘(0 + 𝑦)) = ((𝐹‘0) + (𝐹𝑦))))
10 oveq2 7153 . . . . . . . . . . 11 (𝑦 = 0 → (0 + 𝑦) = (0 + 0))
11 00id 10803 . . . . . . . . . . 11 (0 + 0) = 0
1210, 11syl6eq 2869 . . . . . . . . . 10 (𝑦 = 0 → (0 + 𝑦) = 0)
1312fveq2d 6667 . . . . . . . . 9 (𝑦 = 0 → (𝐹‘(0 + 𝑦)) = (𝐹‘0))
14 fveq2 6663 . . . . . . . . . 10 (𝑦 = 0 → (𝐹𝑦) = (𝐹‘0))
1514oveq2d 7161 . . . . . . . . 9 (𝑦 = 0 → ((𝐹‘0) + (𝐹𝑦)) = ((𝐹‘0) + (𝐹‘0)))
1613, 15eqeq12d 2834 . . . . . . . 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 3572 . . . . . . 7 ((0 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) = ((𝐹‘0) + (𝐹‘0)))
191, 1, 18mp2an 688 . . . . . 6 (𝐹‘0) = ((𝐹‘0) + (𝐹‘0))
205, 19eqtr2i 2842 . . . . 5 ((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0)
214, 4, 1addcani 10821 . . . . 5 (((𝐹‘0) + (𝐹‘0)) = ((𝐹‘0) + 0) ↔ (𝐹‘0) = 0)
2220, 21mpbi 231 . . . 4 (𝐹‘0) = 0
23 sumeq1 15033 . . . . . 6 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = Σ𝑘 ∈ ∅ 𝐵)
24 sum0 15066 . . . . . 6 Σ𝑘 ∈ ∅ 𝐵 = 0
2523, 24syl6eq 2869 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 𝐵 = 0)
2625fveq2d 6667 . . . 4 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = (𝐹‘0))
27 sumeq1 15033 . . . . 5 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = Σ𝑘 ∈ ∅ (𝐹𝐵))
28 sum0 15066 . . . . 5 Σ𝑘 ∈ ∅ (𝐹𝐵) = 0
2927, 28syl6eq 2869 . . . 4 (𝐴 = ∅ → Σ𝑘𝐴 (𝐹𝐵) = 0)
3022, 26, 293eqtr4a 2879 . . 3 (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
3130a1i 11 . 2 (𝜑 → (𝐴 = ∅ → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵)))
32 addcl 10607 . . . . . . . . 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 6871 . . . . . . . . . . 11 (𝜑 → (𝑘𝐴𝐵):𝐴⟶ℂ)
3635adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴𝐵):𝐴⟶ℂ)
37 simprr 769 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
38 f1of 6608 . . . . . . . . . . 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 6524 . . . . . . . . . 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)
4241ffvelrnda 6843 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥) ∈ ℂ)
43 simprl 767 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (♯‘𝐴) ∈ ℕ)
44 nnuz 12269 . . . . . . . . 9 ℕ = (ℤ‘1)
4543, 44eleqtrdi 2920 . . . . . . . 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)))
4739ffvelrnda 6843 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑓𝑥) ∈ 𝐴)
48 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝑘𝐴)
49 eqid 2818 . . . . . . . . . . . . . . . 16 (𝑘𝐴𝐵) = (𝑘𝐴𝐵)
5049fvmpt2 6771 . . . . . . . . . . . . . . 15 ((𝑘𝐴𝐵 ∈ ℂ) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5148, 34, 50syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝑘𝐴𝐵)‘𝑘) = 𝐵)
5251fveq2d 6667 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹𝐵))
53 fvex 6676 . . . . . . . . . . . . . 14 (𝐹𝐵) ∈ V
54 eqid 2818 . . . . . . . . . . . . . . 15 (𝑘𝐴 ↦ (𝐹𝐵)) = (𝑘𝐴 ↦ (𝐹𝐵))
5554fvmpt2 6771 . . . . . . . . . . . . . 14 ((𝑘𝐴 ∧ (𝐹𝐵) ∈ V) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5648, 53, 55sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = (𝐹𝐵))
5752, 56eqtr4d 2856 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5857ralrimiva 3179 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
5958ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ∀𝑘𝐴 (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘))
60 nfcv 2974 . . . . . . . . . . . . 13 𝑘𝐹
61 nffvmpt1 6674 . . . . . . . . . . . . 13 𝑘((𝑘𝐴𝐵)‘(𝑓𝑥))
6260, 61nffv 6673 . . . . . . . . . . . 12 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥)))
63 nffvmpt1 6674 . . . . . . . . . . . 12 𝑘((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
6462, 63nfeq 2988 . . . . . . . . . . 11 𝑘(𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))
65 2fveq3 6668 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → (𝐹‘((𝑘𝐴𝐵)‘𝑘)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
66 fveq2 6663 . . . . . . . . . . . 12 (𝑘 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
6765, 66eqeq12d 2834 . . . . . . . . . . 11 (𝑘 = (𝑓𝑥) → ((𝐹‘((𝑘𝐴𝐵)‘𝑘)) = ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑘) ↔ (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥))))
6864, 67rspc 3608 . . . . . . . . . 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 6753 . . . . . . . . . . 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 6667 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(((𝑘𝐴𝐵) ∘ 𝑓)‘𝑥)) = (𝐹‘((𝑘𝐴𝐵)‘(𝑓𝑥))))
73 fvco3 6753 . . . . . . . . . 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 2863 . . . . . . . 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 13405 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
77 fveq2 6663 . . . . . . . . 9 (𝑚 = (𝑓𝑥) → ((𝑘𝐴𝐵)‘𝑚) = ((𝑘𝐴𝐵)‘(𝑓𝑥)))
7836ffvelrnda 6843 . . . . . . . . 9 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴𝐵)‘𝑚) ∈ ℂ)
7977, 43, 37, 78, 71fsum 15065 . . . . . . . 8 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = (seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴)))
8079fveq2d 6667 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘(seq1( + , ((𝑘𝐴𝐵) ∘ 𝑓))‘(♯‘𝐴))))
81 fveq2 6663 . . . . . . . 8 (𝑚 = (𝑓𝑥) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = ((𝑘𝐴 ↦ (𝐹𝐵))‘(𝑓𝑥)))
822ffvelrni 6842 . . . . . . . . . . . 12 (𝐵 ∈ ℂ → (𝐹𝐵) ∈ ℂ)
8334, 82syl 17 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝐹𝐵) ∈ ℂ)
8483fmpttd 6871 . . . . . . . . . 10 (𝜑 → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8584adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝑘𝐴 ↦ (𝐹𝐵)):𝐴⟶ℂ)
8685ffvelrnda 6843 . . . . . . . 8 (((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) ∧ 𝑚𝐴) → ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) ∈ ℂ)
8781, 43, 37, 86, 74fsum 15065 . . . . . . 7 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = (seq1( + , ((𝑘𝐴 ↦ (𝐹𝐵)) ∘ 𝑓))‘(♯‘𝐴)))
8876, 80, 873eqtr4d 2863 . . . . . 6 ((𝜑 ∧ ((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚))
89 sumfc 15054 . . . . . . 7 Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚) = Σ𝑘𝐴 𝐵
9089fveq2i 6666 . . . . . 6 (𝐹‘Σ𝑚𝐴 ((𝑘𝐴𝐵)‘𝑚)) = (𝐹‘Σ𝑘𝐴 𝐵)
91 sumfc 15054 . . . . . 6 Σ𝑚𝐴 ((𝑘𝐴 ↦ (𝐹𝐵))‘𝑚) = Σ𝑘𝐴 (𝐹𝐵)
9288, 90, 913eqtr3g 2876 . . . . 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 1925 . . 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 15055 . . 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 854 1 (𝜑 → (𝐹‘Σ𝑘𝐴 𝐵) = Σ𝑘𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  c0 4288  cmpt 5137  ccom 5552  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Fincfn 8497  cc 10523  0cc0 10525  1c1 10526   + caddc 10528  cn 11626  cuz 12231  ...cfz 12880  seqcseq 13357  chash 13678  Σcsu 15030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-sum 15031
This theorem is referenced by:  fsumre  15151  fsumim  15152  fsumcj  15153
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