zproddc - Intuitionistic Logic Explorer

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Theorem zproddc 12273

Description: Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)

**Hypotheses**
Ref	Expression
zprod.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
zprod.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
zproddc.3	⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
zprod.4	⊢ (𝜑 → 𝐴 ⊆ 𝑍)
zproddc.dc	⊢ (𝜑 → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
zprod.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
zprod.6	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
zproddc	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Distinct variable groups: 𝐴,𝑗,𝑘,𝑛,𝑦 𝐵,𝑗,𝑛,𝑦 𝑘,𝐹 𝑗,𝑀,𝑘,𝑛,𝑦 𝑗,𝑍,𝑘,𝑛 𝜑,𝑗,𝑘,𝑛,𝑦 Allowed substitution hints: 𝐵(𝑘) 𝐹(𝑦,𝑗,𝑛) 𝑍(𝑦)

Proof of Theorem zproddc Dummy variables 𝑓 𝑔 𝑖 𝑚 𝑝 𝑟 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . . . 9 ⊢ (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → 𝐴 ⊆ (ℤ_≥‘𝑚))
2		simprr 533	. . . . . . . . 9 ⊢ (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)
3	1, 2	jca 306	. . . . . . . 8 ⊢ (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
4		nfcv 2386	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖if(𝑘 ∈ 𝐴, 𝐵, 1)
5		nfv 1577	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑖 ∈ 𝐴
6		nfcsb1v 3173	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵
7		nfcv 2386	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘1
8	5, 6, 7	nfif 3653	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1)
9		eleq1w 2295	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
10		csbeq1a 3149	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑖 → 𝐵 = ⦋𝑖 / 𝑘⦌𝐵)
11	9, 10	ifbieq1d 3647	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑖 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1))
12	4, 8, 11	cbvmpt 4207	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑖 ∈ ℤ ↦ if(𝑖 ∈ 𝐴, ⦋𝑖 / 𝑘⦌𝐵, 1))
13		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝜑)
14		zprod.6	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
15	14	ralrimiva 2617	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
16	6	nfel1 2397	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ
17	10	eleq1d 2303	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝐵 ∈ ℂ ↔ ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
18	16, 17	rspc 2917	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
19	15, 18	syl5 32	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → (𝜑 → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ))
20	13, 19	mpan9 281	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
21		simplr 529	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝑚 ∈ ℤ)
22		zprod.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23	22	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝑀 ∈ ℤ)
24		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝐴 ⊆ (ℤ_≥‘𝑚))
25		zprod.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝑍)
26		zprod.1	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27	25, 26	sseqtrdi 3288	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
28	27	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → 𝐴 ⊆ (ℤ_≥‘𝑀))
29		zproddc.dc	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
30	26	raleqi 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴)
31	29, 30	sylib 122	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴)
32		eleq1w 2295	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝐴 ↔ 𝑖 ∈ 𝐴))
33	32	dcbid 846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑖 ∈ 𝐴))
34	33	cbvralv 2780	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑖 ∈ (ℤ_≥‘𝑀)DECID 𝑖 ∈ 𝐴)
35	31, 34	sylib 122	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑖 ∈ (ℤ_≥‘𝑀)DECID 𝑖 ∈ 𝐴)
36	35	r19.21bi 2632	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
37	36	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
38	37	adantlr 477	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
39	38	adantlr 477	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
40		simp-4l 543	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)) → 𝜑)
41		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)) → ¬ 𝑖 ∈ (ℤ_≥‘𝑀))
42	27	ssneld 3242	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (¬ 𝑖 ∈ (ℤ_≥‘𝑀) → ¬ 𝑖 ∈ 𝐴))
43	40, 41, 42	sylc 62	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)) → ¬ 𝑖 ∈ 𝐴)
44	43	olcd 742	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)) → (𝑖 ∈ 𝐴 ∨ ¬ 𝑖 ∈ 𝐴))
45		df-dc 843	. . . . . . . . . . . . 13 ⊢ (DECID 𝑖 ∈ 𝐴 ↔ (𝑖 ∈ 𝐴 ∨ ¬ 𝑖 ∈ 𝐴))
46	44, 45	sylibr 134	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) ∧ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
47		eluzelz 9869	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (ℤ_≥‘𝑚) → 𝑖 ∈ ℤ)
48		eluzdc 9948	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℤ ∧ 𝑖 ∈ ℤ) → DECID 𝑖 ∈ (ℤ_≥‘𝑀))
49	23, 47, 48	syl2an 289	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → DECID 𝑖 ∈ (ℤ_≥‘𝑀))
50		exmiddc 844	. . . . . . . . . . . . 13 ⊢ (DECID 𝑖 ∈ (ℤ_≥‘𝑀) → (𝑖 ∈ (ℤ_≥‘𝑀) ∨ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)))
51	49, 50	syl 14	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → (𝑖 ∈ (ℤ_≥‘𝑀) ∨ ¬ 𝑖 ∈ (ℤ_≥‘𝑀)))
52	39, 46, 51	mpjaodan 806	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → DECID 𝑖 ∈ 𝐴)
53	12, 20, 21, 23, 24, 28, 52, 38	prodrbdc 12268	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
54	53	biimpd 144	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℤ) ∧ 𝐴 ⊆ (ℤ_≥‘𝑚)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
55	54	expimpd 363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
56	3, 55	syl5 32	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
57	56	rexlimdva 2662	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
58		uzssz 9880	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
59	27, 58	sstrdi 3252	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
60	59	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ⊆ ℤ)
61		1zzd 9609	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 1 ∈ ℤ)
62		nnz 9601	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
63	62	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
64	63	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℤ)
65	61, 64	fzfigd 10800	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ∈ Fin)
66		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
67		f1oeng 6998	. . . . . . . . . . . . . . 15 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
68	65, 66, 67	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
69	68	ensymd 7025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ≈ (1...𝑚))
70		enfii 7131	. . . . . . . . . . . . 13 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
71	65, 69, 70	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝐴 ∈ Fin)
72		zfz1iso 11221	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
73	60, 71, 72	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
74		simpll 527	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
75	74, 19	mpan9 281	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ 𝐴) → ⦋𝑖 / 𝑘⦌𝐵 ∈ ℂ)
76		breq1 4114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
77		fveq2 5672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
78	77	csbeq1d 3147	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
79	76, 78	ifbieq1d 3647	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))
80		csbcow 3151	. . . . . . . . . . . . . . . . . 18 ⊢ ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵
81		ifeq1 3627	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))
82	80, 81	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1)
83	79, 82	eqtr4di 2285	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1))
84	83	cbvmptv 4208	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1))
85		eqid 2234	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑔‘𝑗) / 𝑖⦌⦋𝑖 / 𝑘⦌𝐵, 1))
86	36	ad4ant14 514	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑖 ∈ (ℤ_≥‘𝑀)) → DECID 𝑖 ∈ 𝐴)
87		simplr 529	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
88	22	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑀 ∈ ℤ)
89	27	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ_≥‘𝑀))
90		simprl 531	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
91		simprr 533	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
92	12, 75, 84, 85, 86, 87, 88, 89, 90, 91	prodmodclem2a 12270	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
93	65	adantrr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (1...𝑚) ∈ Fin)
94	93, 90	fihasheqf1od 11160	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = (♯‘𝐴))
95	87	nnnn0d 9558	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ₀)
96		hashfz1 11154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ ℕ₀ → (♯‘(1...𝑚)) = 𝑚)
97	95, 96	syl 14	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘(1...𝑚)) = 𝑚)
98	94, 97	eqtr3d 2269	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (♯‘𝐴) = 𝑚)
99	98	breq2d 4123	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ≤ (♯‘𝐴) ↔ 𝑛 ≤ 𝑚))
100	99	ifbid 3646	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
101	100	mpteq2dv 4203	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
102	101	seqeq3d 10824	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))))
103	102	fveq1d 5674	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
104	92, 103	breqtrd 4137	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
105	104	expr 375	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))
106	105	exlimdv 1868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))
107	73, 106	mpd 13	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
108		breq2 4115	. . . . . . . . . 10 ⊢ (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))
109	107, 108	syl5ibrcom 157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
110	109	expimpd 363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
111	110	exlimdv 1868	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
112	111	rexlimdva 2662	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
113	57, 112	jaod 725	. . . . 5 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
114	22	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝑀 ∈ ℤ)
115	27	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → 𝐴 ⊆ (ℤ_≥‘𝑀))
116	31	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴)
117	115, 116	jca 306	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴))
118		zproddc.3	. . . . . . . . . . 11 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))
119	26	eleq2i 2301	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
120		eluzelz 9869	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
121	120	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑛 ∈ ℤ)
122		simpr 110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → 𝑝 ∈ (ℤ_≥‘𝑛))
123		simplr 529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ (ℤ_≥‘𝑀))
124		uztrn 9877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ∈ (ℤ_≥‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑝 ∈ (ℤ_≥‘𝑀))
125	122, 123, 124	syl2anc 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → 𝑝 ∈ (ℤ_≥‘𝑀))
126	125, 26	eleqtrrdi 2328	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → 𝑝 ∈ 𝑍)
127		zprod.5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
128	127	ralrimiva 2617	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
129	128	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
130		nfv 1577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘 𝑝 ∈ 𝐴
131		nfcsb1v 3173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵
132	130, 131, 7	nfif 3653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)
133	132	nfeq2 2398	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)
134		fveq2 5672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → (𝐹‘𝑘) = (𝐹‘𝑝))
135		eleq1w 2295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑝 → (𝑘 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴))
136		csbeq1a 3149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑝 → 𝐵 = ⦋𝑝 / 𝑘⦌𝐵)
137	135, 136	ifbieq1d 3647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
138	134, 137	eqeq12d 2249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑝 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) ↔ (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)))
139	133, 138	rspc 2917	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) → (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1)))
140	126, 129, 139	sylc 62	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
141		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴)
142	15	ad3antrrr 492	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
143	131	nfel1 2397	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ
144	136	eleq1d 2303	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑝 → (𝐵 ∈ ℂ ↔ ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ))
145	143, 144	rspc 2917	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ))
146	141, 142, 145	sylc 62	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) ∧ 𝑝 ∈ 𝐴) → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ)
147		1cnd 8295	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) ∧ ¬ 𝑝 ∈ 𝐴) → 1 ∈ ℂ)
148		eleq1w 2295	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑝 → (𝑗 ∈ 𝐴 ↔ 𝑝 ∈ 𝐴))
149	148	dcbid 846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑝 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑝 ∈ 𝐴))
150	29	ad2antrr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
151	149, 150, 126	rspcdva 2928	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → DECID 𝑝 ∈ 𝐴)
152	146, 147, 151	ifcldadc 3654	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ)
153	140, 152	eqeltrd 2311	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑝) ∈ ℂ)
154		simpr 110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → 𝑟 ∈ (ℤ_≥‘𝑛))
155		simplr 529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ (ℤ_≥‘𝑀))
156		uztrn 9877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑟 ∈ (ℤ_≥‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → 𝑟 ∈ (ℤ_≥‘𝑀))
157	154, 155, 156	syl2anc 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → 𝑟 ∈ (ℤ_≥‘𝑀))
158	157, 26	eleqtrrdi 2328	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → 𝑟 ∈ 𝑍)
159	128	ad2antrr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
160		nfv 1577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘 𝑟 ∈ 𝐴
161		nfcsb1v 3173	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑟 / 𝑘⦌𝐵
162	160, 161, 7	nfif 3653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)
163	162	nfeq2 2398	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘(𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)
164		fveq2 5672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑟 → (𝐹‘𝑘) = (𝐹‘𝑟))
165		eleq1w 2295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ 𝐴 ↔ 𝑟 ∈ 𝐴))
166		csbeq1a 3149	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑟 → 𝐵 = ⦋𝑟 / 𝑘⦌𝐵)
167	165, 166	ifbieq1d 3647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑟 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
168	164, 167	eqeq12d 2249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑟 → ((𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) ↔ (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)))
169	163, 168	rspc 2917	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1)))
170	158, 159, 169	sylc 62	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
171	58, 157	sselid 3238	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → 𝑟 ∈ ℤ)
172		simpr 110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
173	15	ad3antrrr 492	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
174	161	nfel1 2397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ
175	166	eleq1d 2303	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑟 → (𝐵 ∈ ℂ ↔ ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ))
176	174, 175	rspc 2917	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ))
177	172, 173, 176	sylc 62	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) ∧ 𝑟 ∈ 𝐴) → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ)
178		1cnd 8295	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) ∧ ¬ 𝑟 ∈ 𝐴) → 1 ∈ ℂ)
179		eleq1w 2295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑟 → (𝑗 ∈ 𝐴 ↔ 𝑟 ∈ 𝐴))
180	179	dcbid 846	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑟 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑟 ∈ 𝐴))
181	29	ad2antrr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
182	180, 181, 158	rspcdva 2928	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → DECID 𝑟 ∈ 𝐴)
183	177, 178, 182	ifcldadc 3654	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ)
184		nfcv 2386	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑘𝑟
185		eqid 2234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
186	184, 162, 167, 185	fvmptf 5772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑟 ∈ ℤ ∧ if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
187	171, 183, 186	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
188	170, 187	eqtr4d 2270	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑟) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟))
189		mulcl 8259	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (𝑝 · 𝑞) ∈ ℂ)
190	189	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ)
191	121, 153, 188, 190	seq3feq 10849	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → seq𝑛( · , 𝐹) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))))
192	191	breq1d 4121	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (seq𝑛( · , 𝐹) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
193	192	anbi2d 464	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → ((𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
194	193	exbidv 1874	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
195	119, 194	sylan2b 287	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
196	195	rexbidva 2541	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
197	118, 196	mpbid 147	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
198	26	rexeqi 2748	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
199	197, 198	sylib 122	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
200	199	anim1i 340	. . . . . . . 8 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
201		fveq2 5672	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑀))
202	201	sseq2d 3270	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝐴 ⊆ (ℤ_≥‘𝑚) ↔ 𝐴 ⊆ (ℤ_≥‘𝑀)))
203	201	raleqdv 2749	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴))
204	202, 203	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ↔ (𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴)))
205	201	rexeqdv 2750	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)))
206		seqeq1 10819	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑀 → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))))
207	206	breq1d 4121	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
208	205, 207	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → ((∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)))
209	204, 208	anbi12d 473	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))))
210	209	rspcev 2923	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ ((𝐴 ⊆ (ℤ_≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))) → ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)))
211	114, 117, 200, 210	syl12anc 1272	. . . . . . 7 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)))
212	211	orcd 741	. . . . . 6 ⊢ ((𝜑 ∧ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
213	212	ex 115	. . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))))
214	113, 213	impbid 129	. . . 4 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) ↔ seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
215		eluzelz 9869	. . . . . . . 8 ⊢ (𝑝 ∈ (ℤ_≥‘𝑀) → 𝑝 ∈ ℤ)
216		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴)
217	15	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
218	216, 217, 145	sylc 62	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) ∧ 𝑝 ∈ 𝐴) → ⦋𝑝 / 𝑘⦌𝐵 ∈ ℂ)
219		1cnd 8295	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) ∧ ¬ 𝑝 ∈ 𝐴) → 1 ∈ ℂ)
220	29	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
221	26	eleq2i 2301	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ 𝑍 ↔ 𝑝 ∈ (ℤ_≥‘𝑀))
222	221	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (ℤ_≥‘𝑀) → 𝑝 ∈ 𝑍)
223	222	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → 𝑝 ∈ 𝑍)
224	149, 220, 223	rspcdva 2928	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → DECID 𝑝 ∈ 𝐴)
225	218, 219, 224	ifcldadc 3654	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ)
226		nfcv 2386	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑝
227	226, 132, 137, 185	fvmptf 5772	. . . . . . . 8 ⊢ ((𝑝 ∈ ℤ ∧ if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1) ∈ ℂ) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
228	215, 225, 227	syl2an2 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) = if(𝑝 ∈ 𝐴, ⦋𝑝 / 𝑘⦌𝐵, 1))
229	228, 225	eqeltrd 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑝) ∈ ℂ)
230		eluzelz 9869	. . . . . . . 8 ⊢ (𝑟 ∈ (ℤ_≥‘𝑀) → 𝑟 ∈ ℤ)
231		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → 𝑟 ∈ 𝐴)
232	15	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
233	231, 232, 176	sylc 62	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) ∧ 𝑟 ∈ 𝐴) → ⦋𝑟 / 𝑘⦌𝐵 ∈ ℂ)
234		1cnd 8295	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) ∧ ¬ 𝑟 ∈ 𝐴) → 1 ∈ ℂ)
235	29	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → ∀𝑗 ∈ 𝑍 DECID 𝑗 ∈ 𝐴)
236	26	eleq2i 2301	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝑍 ↔ 𝑟 ∈ (ℤ_≥‘𝑀))
237	236	biimpri 133	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (ℤ_≥‘𝑀) → 𝑟 ∈ 𝑍)
238	237	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → 𝑟 ∈ 𝑍)
239	180, 235, 238	rspcdva 2928	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → DECID 𝑟 ∈ 𝐴)
240	233, 234, 239	ifcldadc 3654	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1) ∈ ℂ)
241	230, 240, 186	syl2an2 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
242	128	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
243	238, 242, 169	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → (𝐹‘𝑟) = if(𝑟 ∈ 𝐴, ⦋𝑟 / 𝑘⦌𝐵, 1))
244	241, 243	eqtr4d 2270	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (ℤ_≥‘𝑀)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑟) = (𝐹‘𝑟))
245	189	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ℂ ∧ 𝑞 ∈ ℂ)) → (𝑝 · 𝑞) ∈ ℂ)
246	22, 229, 244, 245	seq3feq 10849	. . . . 5 ⊢ (𝜑 → seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑀( · , 𝐹))
247	246	breq1d 4121	. . . 4 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
248	214, 247	bitrd 188	. . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) ↔ seq𝑀( · , 𝐹) ⇝ 𝑥))
249	248	iotabidv 5337	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥))
250		df-proddc 12245	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ_≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
251		df-fv 5362	. 2 ⊢ ( ⇝ ‘seq𝑀( · , 𝐹)) = (℩𝑥seq𝑀( · , 𝐹) ⇝ 𝑥)
252	249, 250, 251	3eqtr4g 2292	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∃wex 1541 ∈ wcel 2205 ∀wral 2522 ∃wrex 2523 ⦋csb 3140 ⊆ wss 3213 ifcif 3622 class class class wbr 4111 ↦ cmpt 4173 ℩cio 5312 –1-1-onto→wf1o 5353 ‘cfv 5354 Isom wiso 5355 (class class class)co 6052 ≈ cen 6975 Fincfn 6977 ℂcc 8130 0cc0 8132 1c1 8133 · cmul 8137 < clt 8313 ≤ cle 8314 # cap 8860 ℕcn 9242 ℕ₀cn0 9501 ℤcz 9582 ℤ_≥cuz 9859 ...cfz 10348 seqcseq 10816 ♯chash 11146 ⇝ cli 11971 ∏cprod 12244

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 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250

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 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-irdg 6603 df-frec 6624 df-1o 6649 df-oadd 6653 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-fz 10349 df-fzo 10484 df-seqfrec 10817 df-exp 10908 df-ihash 11147 df-cj 11535 df-rsqrt 11691 df-abs 11692 df-clim 11972 df-proddc 12245

This theorem is referenced by: iprodap 12274 zprodap0 12275 prodssdc 12283

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