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Theorem summo 15062
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 6663 . . . . . . . . . 10 (𝑚 = 𝑛 → (ℤ𝑚) = (ℤ𝑛))
21sseq2d 3996 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑛)))
3 seqeq1 13360 . . . . . . . . . 10 (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
43breq1d 5067 . . . . . . . . 9 (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
52, 4anbi12d 630 . . . . . . . 8 (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
65cbvrexvw 3448 . . . . . . 7 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
7 reeanv 3365 . . . . . . . . 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 776 . . . . . . . . . . . . 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 summo.2 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
1110ad4ant14 748 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
12 simplrl 773 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
13 simplrr 774 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
14 simprll 775 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑚))
15 simprrl 777 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ𝑛))
169, 11, 12, 13, 14, 15sumrb 15058 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑛( + , 𝐹) ⇝ 𝑥))
178, 16mpbid 233 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
18 simprrr 778 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
19 climuni 14897 . . . . . . . . . . . 12 ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
2017, 18, 19syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
2120exp31 420 . . . . . . . . . 10 (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
2221rexlimdvv 3290 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
237, 22syl5bir 244 . . . . . . . 8 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
2423expdimp 453 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ𝑛) ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
256, 24syl5bi 243 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
26 summo.3 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
279, 10, 26summolem2 15061 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
2825, 27jaod 853 . . . . 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))
299, 10, 26summolem2 15061 . . . . . . . 8 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑦 = 𝑥))
30 equcom 2016 . . . . . . . 8 (𝑦 = 𝑥𝑥 = 𝑦)
3129, 30syl6ib 252 . . . . . . 7 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
3231impancom 452 . . . . . 6 ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
33 oveq2 7153 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
3433f1oeq2d 6604 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑛)–1-1-onto𝐴))
35 fveq2 6663 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
3635eqeq2d 2829 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
3734, 36anbi12d 630 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
3837exbidv 1913 . . . . . . . . 9 (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛))))
39 f1oeq1 6597 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴))
40 fveq1 6662 . . . . . . . . . . . . . . . . 17 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
4140csbeq1d 3884 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑔(𝑓𝑛) / 𝑘𝐵 = (𝑔𝑛) / 𝑘𝐵)
4241mpteq2dv 5153 . . . . . . . . . . . . . . 15 (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))
4326, 42syl5eq 2865 . . . . . . . . . . . . . 14 (𝑓 = 𝑔𝐺 = (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))
4443seqeq3d 13365 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵)))
4544fveq1d 6665 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))
4645eqeq2d 2829 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
4739, 46anbi12d 630 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ (𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛))))
4847cbvexvw 2035 . . . . . . . . 9 (∃𝑓(𝑓:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto𝐴𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
4938, 48syl6bb 288 . . . . . . . 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))))
5049cbvrexvw 3448 . . . . . . 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)))
51 reeanv 3365 . . . . . . . . 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))))
52 exdistrv 1947 . . . . . . . . . . 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))))
53 an4 652 . . . . . . . . . . . . 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))))
5410ad4ant14 748 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
55 fveq2 6663 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑗 → (𝑓𝑛) = (𝑓𝑗))
5655csbeq1d 3884 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑗) / 𝑘𝐵)
5756cbvmptv 5160 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
5826, 57eqtri 2841 . . . . . . . . . . . . . . . 16 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
59 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑗 → (𝑔𝑛) = (𝑔𝑗))
6059csbeq1d 3884 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗(𝑔𝑛) / 𝑘𝐵 = (𝑔𝑗) / 𝑘𝐵)
6160cbvmptv 5160 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵) = (𝑗 ∈ ℕ ↦ (𝑔𝑗) / 𝑘𝐵)
62 simplr 765 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ))
63 simprl 767 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
64 simprr 769 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto𝐴𝑔:(1...𝑛)–1-1-onto𝐴)) → 𝑔:(1...𝑛)–1-1-onto𝐴)
659, 54, 58, 61, 62, 63, 64summolem3 15059 . . . . . . . . . . . . . . 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))
66 eqeq12 2832 . . . . . . . . . . . . . . 15 ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)) → (𝑥 = 𝑦 ↔ (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑔𝑛) / 𝑘𝐵))‘𝑛)))
6765, 66syl5ibrcom 248 . . . . . . . . . . . . . 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))
6867expimpd 454 . . . . . . . . . . . . 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))
6953, 68syl5bi 243 . . . . . . . . . . . 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))
7069exlimdvv 1926 . . . . . . . . . . 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))
7152, 70syl5bir 244 . . . . . . . . . 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))
7271rexlimdvva 3291 . . . . . . . . 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))
7351, 72syl5bir 244 . . . . . . . 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))
7473expdimp 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))
7550, 74syl5bi 243 . . . . . 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))
7632, 75jaod 853 . . . . 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))
7728, 76jaodan 951 . . . 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))
7877expimpd 454 . . 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))
7978alrimivv 1920 . 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))
80 breq2 5061 . . . . . 6 (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
8180anbi2d 628 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
8281rexbidv 3294 . . . 4 (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
83 eqeq1 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
8483anbi2d 628 . . . . . 6 (𝑥 = 𝑦 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
8584exbidv 1913 . . . . 5 (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺)‘𝑚))))
8685rexbidv 3294 . . . 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))))
8782, 86orbi12d 912 . . 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)))))
8887mo4 2643 . 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))
8979, 88sylibr 235 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  wa 396  wo 841  wal 1526   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  wrex 3136  csb 3880  wss 3933  ifcif 4463   class class class wbr 5057  cmpt 5137  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525  1c1 10526   + caddc 10528  cn 11626  cz 11969  cuz 12231  ...cfz 12880  seqcseq 13357  cli 14829
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-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
This theorem is referenced by:  fsum  15065
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