Theorem prodmodclem2 11723

Description: Lemma for prodmodc 11724. (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 1179	. . 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 2591	. 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 5555	. . . . . 6 ⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) = (ℤ≥‘𝑤))
7	6	sseq2d 3210	. . . . 5 ⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤)))
8	6	raleqdv 2696	. . . . 5 ⊢ (𝑚 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴))
9		seqeq1 10524	. . . . . 6 ⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
10	9	breq1d 4040	. . . . 5 ⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
11	7, 8, 10	3anbi123d 1323	. . . 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 2731	. . 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 2664	. . . . 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 1046	. . . . . . . . . . . 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 1044	. . . . . . . . . . . . . . 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 9615	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑤) ⊆ ℤ
17	15, 16	sstrdi 3192	. . . . . . . . . . . . . 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 9347	. . . . . . . . . . . . . . . 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 9441	. . . . . . . . . . . . . . . 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 10505	. . . . . . . . . . . . . . 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 6813	. . . . . . . . . . . . . . . . 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 6839	. . . . . . . . . . . . . . 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 6932	. . . . . . . . . . . . . . 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 10915	. . . . . . . . . . . . . 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 2193	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1))
35		simpll2 1039	. . . . . . . . . . . . . . . . . 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 2254	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
38	37	dcbid 839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
39	38	rspcv 2861	. . . . . . . . . . . . . . . . 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 11722	. . . . . . . . . . . . . . 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 1830	. . . . . . . . . . . . 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 11439	. . . . . . . . . . . 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 2203	. . . . . . . . . . 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 1830	. . . . . . 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 2619	. . . . 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 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ⦋csb 3081   ⊆ wss 3154  ifcif 3558   class class class wbr 4030   ↦ cmpt 4091  –1-1-onto→wf1o 5254  ‘cfv 5255   Isom wiso 5256  (class class class)co 5919   ≈ cen 6794  Fincfn 6796  ℂcc 7872  0cc0 7874  1c1 7875   · cmul 7879   < clt 8056   ≤ cle 8057   # cap 8602  ℕcn 8984  ℤcz 9320  ℤ_≥cuz 9595  ...cfz 10077  seqcseq 10521  ♯chash 10849   ⇝ cli 11424

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425

This theorem is referenced by:  prodmodc  11724