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Theorem isummolem2 10736
 Description: Lemma for isummo 10737. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 8-Sep-2022.)
Hypotheses
Ref Expression
isummo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
isummo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
isummolem2.g 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
Assertion
Ref Expression
isummolem2 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹, ℂ) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem isummolem2
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5289 . . . . 5 (𝑚 = 𝑎 → (ℤ𝑚) = (ℤ𝑎))
21sseq2d 3052 . . . 4 (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑎)))
31raleqdv 2568 . . . 4 (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴))
4 iseqeq1 9823 . . . . 5 (𝑚 = 𝑎 → seq𝑚( + , 𝐹, ℂ) = seq𝑎( + , 𝐹, ℂ))
54breq1d 3847 . . . 4 (𝑚 = 𝑎 → (seq𝑚( + , 𝐹, ℂ) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥))
62, 3, 53anbi123d 1248 . . 3 (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹, ℂ) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)))
76cbvrexv 2591 . 2 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹, ℂ) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥))
8 simplr3 987 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)
9 simplr1 985 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑎))
10 uzssz 9007 . . . . . . . . . . . 12 (ℤ𝑎) ⊆ ℤ
119, 10syl6ss 3035 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℤ)
12 1zzd 8747 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 1 ∈ ℤ)
13 simprl 498 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℕ)
1413nnzd 8837 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℤ)
1512, 14fzfigd 9803 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ∈ Fin)
16 simprr 499 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
17 f1oeng 6454 . . . . . . . . . . . . . 14 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
1815, 16, 17syl2anc 403 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
1918ensymd 6480 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
20 enfii 6570 . . . . . . . . . . . 12 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
2115, 19, 20syl2anc 403 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ∈ Fin)
22 zfz1iso 10211 . . . . . . . . . . 11 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
2311, 21, 22syl2anc 403 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24 isummo.1 . . . . . . . . . . . . 13 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
25 simplll 500 . . . . . . . . . . . . . 14 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26 isummo.2 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
2725, 26sylan 277 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
28 eleq1w 2148 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
2928dcbid 786 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
30 simpr2 950 . . . . . . . . . . . . . . 15 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
3130ad2antrr 472 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴)
32 simpr 108 . . . . . . . . . . . . . 14 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → 𝑘 ∈ (ℤ𝑎))
3329, 31, 32rspcdva 2727 . . . . . . . . . . . . 13 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑎)) → DECID 𝑘𝐴)
34 isummolem2.g . . . . . . . . . . . . 13 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), (𝑓𝑛) / 𝑘𝐵, 0))
35 eqid 2088 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑔𝑛) / 𝑘𝐵, 0))
36 simprll 504 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37 simpllr 501 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38 simplr1 985 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑎))
39 simprlr 505 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
40 simprr 499 . . . . . . . . . . . . 13 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4124, 27, 33, 34, 35, 36, 37, 38, 39, 40isummolem2a 10735 . . . . . . . . . . . 12 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑚))
4241expr 367 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑚)))
4342exlimdv 1747 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑚)))
4423, 43mpd 13 . . . . . . . . 9 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑎( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑚))
45 climuni 10645 . . . . . . . . 9 ((seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = (seq1( + , 𝐺, ℂ)‘𝑚))
468, 44, 45syl2anc 403 . . . . . . . 8 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( + , 𝐺, ℂ)‘𝑚))
4746anassrs 392 . . . . . . 7 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑥 = (seq1( + , 𝐺, ℂ)‘𝑚))
48 eqeq2 2097 . . . . . . 7 (𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚) → (𝑥 = 𝑦𝑥 = (seq1( + , 𝐺, ℂ)‘𝑚)))
4947, 48syl5ibrcom 155 . . . . . 6 (((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚) → 𝑥 = 𝑦))
5049expimpd 355 . . . . 5 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
5150exlimdv 1747 . . . 4 ((((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
5251rexlimdva 2489 . . 3 (((𝜑𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
5352r19.29an 2510 . 2 ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ𝑎) ∧ ∀𝑗 ∈ (ℤ𝑎)DECID 𝑗𝐴 ∧ seq𝑎( + , 𝐹, ℂ) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
547, 53sylan2b 281 1 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , 𝐹, ℂ) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( + , 𝐺, ℂ)‘𝑚)) → 𝑥 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  DECID wdc 780   ∧ w3a 924   = wceq 1289  ∃wex 1426   ∈ wcel 1438  ∀wral 2359  ∃wrex 2360  ⦋csb 2931   ⊆ wss 2997  ifcif 3389   class class class wbr 3837   ↦ cmpt 3891  –1-1-onto→wf1o 5001  ‘cfv 5002   Isom wiso 5003  (class class class)co 5634   ≈ cen 6435  Fincfn 6437  ℂcc 7327  0cc0 7329  1c1 7330   + caddc 7332   < clt 7501   ≤ cle 7502  ℕcn 8394  ℤcz 8720  ℤ≥cuz 8988  ...cfz 9393  seqcseq4 9816  ♯chash 10148   ⇝ cli 10630 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444 This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-isom 5011  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-frec 6138  df-1o 6163  df-oadd 6167  df-er 6272  df-en 6438  df-dom 6439  df-fin 6440  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-iseq 9818  df-seq3 9819  df-exp 9920  df-ihash 10149  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631 This theorem is referenced by:  isummo  10737
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