Theorem prodmodclem2 11598

Description: Lemma for prodmodc 11599. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)

Hypotheses

Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmodc.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))

Assertion

Ref	Expression
prodmodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (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.

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → 𝐴 ⊆ (ℤ≥‘𝑚))
2		simplr 528	. . . 4 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
3		simprr 531	. . . 4 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → seq𝑚( · , 𝐹) ⇝ 𝑥)
4	1, 2, 3	3jca 1178	. . 3 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
5	4	reximi 2584	. 2 ⊢ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))
6		fveq2 5527	. . . . . 6 ⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) = (ℤ≥‘𝑤))
7	6	sseq2d 3197	. . . . 5 ⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤)))
8	6	raleqdv 2689	. . . . 5 ⊢ (𝑚 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴))
9		seqeq1 10461	. . . . . 6 ⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
10	9	breq1d 4025	. . . . 5 ⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
11	7, 8, 10	3anbi123d 1322	. . . 4 ⊢ (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)))
12	11	cbvrexvw 2720	. . 3 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥))
13		reeanv 2657	. . . . 5 ⊢ (∃𝑤 ∈ ℤ ∃𝑚 ∈ ℕ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))))
14		simprl3 1045	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑤( · , 𝐹) ⇝ 𝑥)
15		simprl1 1043	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑤))
16		uzssz 9560	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑤) ⊆ ℤ
17	15, 16	sstrdi 3179	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
18		1zzd 9293	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
19		simplrr 536	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
20	19	nnzd 9387	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
21	18, 20	fzfigd 10444	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
22		simprr 531	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
23		f1oeng 6770	. . . . . . . . . . . . . . . . 17 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
24	21, 22, 23	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
25	24	ensymd 6796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
26		enfii 6887	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
27	21, 25, 26	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
28		zfz1iso 10834	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
29	17, 27, 28	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
30		prodmo.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
31		prodmo.2	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
32	31	ad4ant14 514	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
33		prodmodc.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))
34		eqid 2187	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1))
35		simpll2 1038	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴)) → ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴)
36	35	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴)
37		eleq1w 2248	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
38	37	dcbid 839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
39	38	rspcv 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘𝑤) → (∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 → DECID 𝑘 ∈ 𝐴))
40	36, 39	mpan9 281	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑤)) → DECID 𝑘 ∈ 𝐴)
41		simplrr 536	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
42		simplrl 535	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑤 ∈ ℤ)
43	15	adantrr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑤))
44		simprlr 538	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
45		simprr 531	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
46	30, 32, 33, 34, 40, 41, 42, 43, 44, 45	prodmodclem2a 11597	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))
47	46	expr 375	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)))
48	47	exlimdv 1829	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)))
49	29, 48	mpd 13	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))
50		climuni 11314	. . . . . . . . . . . 12 ⊢ ((seq𝑤( · , 𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)) → 𝑥 = (seq1( · , 𝐺)‘𝑚))
51	14, 49, 50	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( · , 𝐺)‘𝑚))
52		eqeq2 2197	. . . . . . . . . . 11 ⊢ (𝑧 = (seq1( · , 𝐺)‘𝑚) → (𝑥 = 𝑧 ↔ 𝑥 = (seq1( · , 𝐺)‘𝑚)))
53	51, 52	syl5ibrcom 157	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧))
54	53	expr 375	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (𝑓:(1...𝑚)–1-1-onto→𝐴 → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧)))
55	54	impd 254	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
56	55	exlimdv 1829	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
57	56	expimpd 363	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) → (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
58	57	rexlimdvva 2612	. . . . 5 ⊢ (𝜑 → (∃𝑤 ∈ ℤ ∃𝑚 ∈ ℕ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
59	13, 58	biimtrrid 153	. . . 4 ⊢ (𝜑 → ((∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧))
60	59	expdimp 259	. . 3 ⊢ ((𝜑 ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴 ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
61	12, 60	sylan2b 287	. 2 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
62	5, 61	sylan2 286	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 835   ∧ w3a 979   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466  ⦋csb 3069   ⊆ wss 3141  ifcif 3546   class class class wbr 4015   ↦ cmpt 4076  –1-1-onto→wf1o 5227  ‘cfv 5228   Isom wiso 5229  (class class class)co 5888   ≈ cen 6751  Fincfn 6753  ℂcc 7822  0cc0 7824  1c1 7825   · cmul 7829   < clt 8005   ≤ cle 8006   # cap 8551  ℕcn 8932  ℤcz 9266  ℤ_≥cuz 9541  ...cfz 10021  seqcseq 10458  ♯chash 10768   ⇝ cli 11299

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-en 6754  df-dom 6755  df-fin 6756  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-ihash 10769  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300

This theorem is referenced by:  prodmodc  11599