Theorem prodmodclem2 12288

Description: Lemma for prodmodc 12289. (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 529	. . . 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 533	. . . 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 1204	. . 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 2641	. 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 5675	. . . . . 6 ⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) = (ℤ≥‘𝑤))
7	6	sseq2d 3272	. . . . 5 ⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤)))
8	6	raleqdv 2749	. . . . 5 ⊢ (𝑚 = 𝑤 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑤)DECID 𝑗 ∈ 𝐴))
9		seqeq1 10836	. . . . . 6 ⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹))
10	9	breq1d 4124	. . . . 5 ⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥))
11	7, 8, 10	3anbi123d 1349	. . . 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 2785	. . 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 2715	. . . . 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 1071	. . . . . . . . . . . 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 1069	. . . . . . . . . . . . . . 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 9892	. . . . . . . . . . . . . . 15 ⊢ (ℤ≥‘𝑤) ⊆ ℤ
17	15, 16	sstrdi 3254	. . . . . . . . . . . . . 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 9621	. . . . . . . . . . . . . . . 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 538	. . . . . . . . . . . . . . . . 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 9717	. . . . . . . . . . . . . . . 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 10817	. . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . . . 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 7009	. . . . . . . . . . . . . . . . 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 7036	. . . . . . . . . . . . . . 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 7142	. . . . . . . . . . . . . . 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 11238	. . . . . . . . . . . . . 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 2234	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 1))
35		simpll2 1064	. . . . . . . . . . . . . . . . . 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 2295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
38	37	dcbid 846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
39	38	rspcv 2919	. . . . . . . . . . . . . . . . 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 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))) → 𝑚 ∈ ℕ)
42		simplrl 537	. . . . . . . . . . . . . . . 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 540	. . . . . . . . . . . . . . . 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 533	. . . . . . . . . . . . . . . 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 12287	. . . . . . . . . . . . . . 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 1868	. . . . . . . . . . . . 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 12003	. . . . . . . . . . . 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 2244	. . . . . . . . . . 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 1868	. . . . . . 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 2670	. . . . 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 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ⦋csb 3141   ⊆ wss 3214  ifcif 3624   class class class wbr 4114   ↦ cmpt 4176  –1-1-onto→wf1o 5356  ‘cfv 5357   Isom wiso 5358  (class class class)co 6058   ≈ cen 6986  Fincfn 6988  ℂcc 8141  0cc0 8143  1c1 8144   · cmul 8148   < clt 8324   ≤ cle 8325   # cap 8872  ℕcn 9254  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361  seqcseq 10833  ♯chash 11163   ⇝ cli 11988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989

This theorem is referenced by:  prodmodc  12289