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Theorem summo 14389
Description: A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
summo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
summo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summo.3 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
Assertion
Ref Expression
summo (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
Distinct variable groups:   𝑓,𝑘,𝑚,𝑛,𝑥,𝐴   𝑓,𝐹,𝑘,𝑚,𝑛,𝑥   𝑘,𝐺,𝑚,𝑛,𝑥   𝜑,𝑘,𝑚,𝑛   𝐵,𝑓,𝑚,𝑛,𝑥   𝜑,𝑥,𝑓
Allowed substitution hints:   𝐵(𝑘)   𝐺(𝑓)

Proof of Theorem summo
Dummy variables 𝑔 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6153 . . . . . . . . . 10 (𝑚 = 𝑛 → (ℤ𝑚) = (ℤ𝑛))
21sseq2d 3617 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑛)))
3 seqeq1 12752 . . . . . . . . . 10 (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
43breq1d 4628 . . . . . . . . 9 (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
52, 4anbi12d 746 . . . . . . . 8 (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
65cbvrexv 3163 . . . . . . 7 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
7 reeanv 3100 . . . . . . . . 9 (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
8 simprlr 802 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑚( + , 𝐹) ⇝ 𝑥)
9 summo.1 . . . . . . . . . . . . . 14 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
10 simpll 789 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝜑)
11 summo.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1210, 11sylan 488 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
13 simplrl 799 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
14 simplrr 800 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
15 simprll 801 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑚))
16 simprrl 803 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑛))
179, 12, 13, 14, 15, 16sumrb 14385 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑛( + , 𝐹) ⇝ 𝑥))
188, 17mpbid 222 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
19 simprrr 804 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
20 climuni 14225 . . . . . . . . . . . 12 ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
2118, 19, 20syl2anc 692 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
2221exp31 629 . . . . . . . . . 10 (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
2322rexlimdvv 3031 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
247, 23syl5bir 233 . . . . . . . 8 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
2524expdimp 453 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
266, 25syl5bi 232 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
27 summo.3 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
289, 11, 27summolem2 14388 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
2926, 28jaod 395 . . . . 5 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
309, 11, 27summolem2 14388 . . . . . . . 8 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑦 = 𝑥))
31 equcom 1942 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
3230, 31syl6ib 241 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
3332impancom 456 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
34 oveq2 6618 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
35 f1oeq2 6090 . . . . . . . . . . . 12 ((1...𝑚) = (1...𝑛) → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑛)–1-1-onto𝐴))
3634, 35syl 17 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑛)–1-1-onto𝐴))
37 fveq2 6153 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
3837eqeq2d 2631 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
3936, 38anbi12d 746 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
4039exbidv 1847 . . . . . . . . 9 (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
41 f1oeq1 6089 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴))
42 fveq1 6152 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
4342csbeq1d 3525 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔(𝑓𝑛) / 𝑘𝐵 = (𝑔𝑛) / 𝑘𝐵)
4443mpteq2dv 4710 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))
4527, 44syl5eq 2667 . . . . . . . . . . . . . 14 (𝑓 = 𝑔𝐺 = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))
4645seqeq3d 12757 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵)))
4746fveq1d 6155 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))
4847eqeq2d 2631 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
4941, 48anbi12d 746 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
5049cbvexv 2274 . . . . . . . . 9 (∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
5140, 50syl6bb 276 . . . . . . . 8 (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
5251cbvrexv 3163 . . . . . . 7 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
53 reeanv 3100 . . . . . . . . 9 (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
54 eeanv 2181 . . . . . . . . . . 11 (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
55 an4 864 . . . . . . . . . . . . 13 (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) ↔ ((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
56 simpll 789 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝜑)
5756, 11sylan 488 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
58 fveq2 6153 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
5958csbeq1d 3525 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
6059cbvmptv 4715 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
6127, 60eqtri 2643 . . . . . . . . . . . . . . . 16 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
62 fveq2 6153 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
6362csbeq1d 3525 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗(𝑔𝑛) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
6463cbvmptv 4715 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵)
65 simplr 791 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ))
66 simprl 793 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
67 simprr 795 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑔:(1...𝑛)–1-1-onto𝐴)
689, 57, 61, 64, 65, 66, 67summolem3 14386 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))
69 eqeq12 2634 . . . . . . . . . . . . . . 15 ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)) → (𝑥 = 𝑦 ↔ (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
7068, 69syl5ibrcom 237 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)) → 𝑥 = 𝑦))
7170expimpd 628 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7255, 71syl5bi 232 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7372exlimdvv 1859 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (∃𝑓𝑔((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7454, 73syl5bir 233 . . . . . . . . . 10 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → ((∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7574rexlimdvva 3032 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7653, 75syl5bir 233 . . . . . . . 8 (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))) → 𝑥 = 𝑦))
7776expdimp 453 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)) → 𝑥 = 𝑦))
7852, 77syl5bi 232 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
7933, 78jaod 395 . . . . 5 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
8029, 79jaodan 825 . . . 4 ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
8180expimpd 628 . . 3 (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
8281alrimivv 1853 . 2 (𝜑 → ∀𝑥𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
83 breq2 4622 . . . . . 6 (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
8483anbi2d 739 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
8584rexbidv 3046 . . . 4 (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
86 eqeq1 2625 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
8786anbi2d 739 . . . . . 6 (𝑥 = 𝑦 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
8887exbidv 1847 . . . . 5 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
8988rexbidv 3046 . . . 4 (𝑥 = 𝑦 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
9085, 89orbi12d 745 . . 3 (𝑥 = 𝑦 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))))
9190mo4 2516 . 2 (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ ∀𝑥𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
9282, 91sylibr 224 1 (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  ∃*wmo 2470  wrex 2908  csb 3518  wss 3559  ifcif 4063   class class class wbr 4618  cmpt 4678  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cc 9886  0cc0 9888  1c1 9889   + caddc 9891  cn 10972  cz 11329  cuz 11639  ...cfz 12276  seqcseq 12749  cli 14157
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  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 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-sup 8300  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-rp 11785  df-fz 12277  df-fzo 12415  df-seq 12750  df-exp 12809  df-hash 13066  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-clim 14161
This theorem is referenced by:  fsum  14392
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