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Theorem prodmodclem2 11551
Description: Lemma for prodmodc 11552. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
Hypotheses
Ref Expression
prodmo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
prodmo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
prodmodc.3 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))
Assertion
Ref Expression
prodmodclem2 ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
Distinct variable groups:   𝐴,𝑓,𝑗,𝑘,𝑚   𝐵,𝑗   𝑓,𝐹,𝑘,𝑚   𝑗,𝐺   𝜑,𝑓,𝑘,𝑚   𝑥,𝑓,𝑘,𝑚   𝑧,𝑓,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑗,𝑛)   𝐴(𝑥,𝑦,𝑧,𝑛)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑘,𝑚,𝑛)   𝐹(𝑥,𝑦,𝑧,𝑗,𝑛)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑘,𝑚,𝑛)

Proof of Theorem prodmodclem2
Dummy variables 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . . 4 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → 𝐴 ⊆ (ℤ𝑚))
2 simplr 528 . . . 4 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
3 simprr 531 . . . 4 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → seq𝑚( · , 𝐹) ⇝ 𝑥)
41, 2, 33jca 1177 . . 3 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
54reximi 2572 . 2 (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
6 fveq2 5507 . . . . . 6 (𝑚 = 𝑤 → (ℤ𝑚) = (ℤ𝑤))
76sseq2d 3183 . . . . 5 (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑤)))
86raleqdv 2676 . . . . 5 (𝑚 = 𝑤 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴))
9 seqeq1 10416 . . . . . 6 (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
109breq1d 4008 . . . . 5 (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
117, 8, 103anbi123d 1312 . . . 4 (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)))
1211cbvrexvw 2706 . . 3 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥))
13 reeanv 2644 . . . . 5 (∃𝑤 ∈ ℤ ∃𝑚 ∈ ℕ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))))
14 simprl3 1044 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑤( · , 𝐹) ⇝ 𝑥)
15 simprl1 1042 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ (ℤ𝑤))
16 uzssz 9518 . . . . . . . . . . . . . . 15 (ℤ𝑤) ⊆ ℤ
1715, 16sstrdi 3165 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ⊆ ℤ)
18 1zzd 9251 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 1 ∈ ℤ)
19 simplrr 536 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℕ)
2019nnzd 9345 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑚 ∈ ℤ)
2118, 20fzfigd 10399 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ∈ Fin)
22 simprr 531 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑓:(1...𝑚)–1-1-onto𝐴)
23 f1oeng 6747 . . . . . . . . . . . . . . . . 17 (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (1...𝑚) ≈ 𝐴)
2421, 22, 23syl2anc 411 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (1...𝑚) ≈ 𝐴)
2524ensymd 6773 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ≈ (1...𝑚))
26 enfii 6864 . . . . . . . . . . . . . . 15 (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
2721, 25, 26syl2anc 411 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝐴 ∈ Fin)
28 zfz1iso 10787 . . . . . . . . . . . . . 14 ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
2917, 27, 28syl2anc 411 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
30 prodmo.1 . . . . . . . . . . . . . . . 16 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
31 prodmo.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
3231ad4ant14 514 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘𝐴) → 𝐵 ∈ ℂ)
33 prodmodc.3 . . . . . . . . . . . . . . . 16 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))
34 eqid 2175 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑔𝑗) / 𝑘𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑔𝑗) / 𝑘𝐵, 1))
35 simpll2 1037 . . . . . . . . . . . . . . . . . 18 ((((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴)) → ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴)
3635adantl 277 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴)
37 eleq1w 2236 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
3837dcbid 838 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑘 → (DECID 𝑗𝐴DECID 𝑘𝐴))
3938rspcv 2835 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ𝑤) → (∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴DECID 𝑘𝐴))
4036, 39mpan9 281 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ𝑤)) → DECID 𝑘𝐴)
41 simplrr 536 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
42 simplrl 535 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑤 ∈ ℤ)
4315adantrr 479 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ𝑤))
44 simprlr 538 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto𝐴)
45 simprr 531 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
4630, 32, 33, 34, 40, 41, 42, 43, 44, 45prodmodclem2a 11550 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))
4746expr 375 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)))
4847exlimdv 1817 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)))
4929, 48mpd 13 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))
50 climuni 11267 . . . . . . . . . . . 12 ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)) → 𝑥 = (seq1( · , 𝐺)‘𝑚))
5114, 49, 50syl2anc 411 . . . . . . . . . . 11 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → 𝑥 = (seq1( · , 𝐺)‘𝑚))
52 eqeq2 2185 . . . . . . . . . . 11 (𝑧 = (seq1( · , 𝐺)‘𝑚) → (𝑥 = 𝑧𝑥 = (seq1( · , 𝐺)‘𝑚)))
5351, 52syl5ibrcom 157 . . . . . . . . . 10 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴)) → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧))
5453expr 375 . . . . . . . . 9 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (𝑓:(1...𝑚)–1-1-onto𝐴 → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧)))
5554impd 254 . . . . . . . 8 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
5655exlimdv 1817 . . . . . . 7 (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
5756expimpd 363 . . . . . 6 ((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) → (((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
5857rexlimdvva 2600 . . . . 5 (𝜑 → (∃𝑤 ∈ ℤ ∃𝑚 ∈ ℕ ((𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
5913, 58syl5bir 153 . . . 4 (𝜑 → ((∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
6059expdimp 259 . . 3 ((𝜑 ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ𝑤) ∧ ∀𝑗 ∈ (ℤ𝑤)DECID 𝑗𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
6112, 60sylan2b 287 . 2 ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
625, 61sylan2 286 1 ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834  w3a 978   = wceq 1353  wex 1490  wcel 2146  wral 2453  wrex 2454  csb 3055  wss 3127  ifcif 3532   class class class wbr 3998  cmpt 4059  1-1-ontowf1o 5207  cfv 5208   Isom wiso 5209  (class class class)co 5865  cen 6728  Fincfn 6730  cc 7784  0cc0 7786  1c1 7787   · cmul 7791   < clt 7966  cle 7967   # cap 8512  cn 8890  cz 9224  cuz 9499  ...cfz 9977  seqcseq 10413  chash 10721  cli 11252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-frec 6382  df-1o 6407  df-oadd 6411  df-er 6525  df-en 6731  df-dom 6732  df-fin 6733  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-seqfrec 10414  df-exp 10488  df-ihash 10722  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-clim 11253
This theorem is referenced by:  prodmodc  11552
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