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Theorem gexexlem 19875

Description: Lemma for gexex 19876. (Contributed by Mario Carneiro, 24-Apr-2016.)

**Hypotheses**
Ref	Expression
gexex.1	⊢ 𝑋 = (Base‘𝐺)
gexex.2	⊢ 𝐸 = (gEx‘𝐺)
gexex.3	⊢ 𝑂 = (od‘𝐺)
gexexlem.1	⊢ (𝜑 → 𝐺 ∈ Abel)
gexexlem.2	⊢ (𝜑 → 𝐸 ∈ ℕ)
gexexlem.3	⊢ (𝜑 → 𝐴 ∈ 𝑋)
gexexlem.4	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝐴))

**Assertion**
Ref	Expression
gexexlem	⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)

Distinct variable groups: 𝑦,𝐴 𝑦,𝐸 𝑦,𝐺 𝑦,𝑂 𝜑,𝑦 𝑦,𝑋

Proof of Theorem gexexlem Dummy variables 𝑥 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexexlem.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		gexex.1	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
3		gexex.3	. . . 4 ⊢ 𝑂 = (od‘𝐺)
4	2, 3	odcl 19559	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
5	1, 4	syl 17	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ₀)
6		gexexlem.2	. . 3 ⊢ (𝜑 → 𝐸 ∈ ℕ)
7	6	nnnn0d 12539	. 2 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
8		gexexlem.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
9		ablgrp 19808	. . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
11		gexex.2	. . . 4 ⊢ 𝐸 = (gEx‘𝐺)
12	2, 11, 3	gexod 19609	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸)
13	10, 1, 12	syl2anc 593	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∥ 𝐸)
14	8	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Abel)
15	10	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Grp)
16		prmnn 16691	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
17	16	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
18		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
19	6	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐸 ∈ ℕ)
20	1	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ 𝑋)
21	2, 11, 3	gexnnod 19611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ)
22	15, 19, 20, 21	syl3anc 1389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℕ)
23	18, 22	pccld 16869	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℕ₀)
24	17, 23	nnexpcld 14255	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ)
25	24	nnzd 12591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℤ)
26		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
27	2, 26	mulgcl 19116	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℤ ∧ 𝐴 ∈ 𝑋) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋)
28	15, 25, 20, 27	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋)
29		simplr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ 𝑋)
30	2, 11, 3	gexnnod 19611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ)
31	15, 19, 29, 30	syl3anc 1389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℕ)
32		pcdvds 16883	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝑥) ∈ ℕ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥))
33	18, 31, 32	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥))
34	18, 31	pccld 16869	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℕ₀)
35	17, 34	nnexpcld 14255	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ)
36		nndivdvds 16278	. . . . . . . . . . . . . . . 16 ⊢ (((𝑂‘𝑥) ∈ ℕ ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥) ↔ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ))
37	31, 35, 36	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥) ↔ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ))
38	33, 37	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ)
39	38	nnzd 12591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ)
40	2, 26	mulgcl 19116	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Grp ∧ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ ∧ 𝑥 ∈ 𝑋) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋)
41	15, 39, 29, 40	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋)
42	2, 3, 26	odmulg 19579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℤ) → (𝑂‘𝐴) = (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))))
43	15, 20, 25, 42	syl3anc 1389	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) = (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))))
44		pcdvds 16883	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴))
45	18, 22, 44	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴))
46		gcdeq 16570	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ ∧ (𝑂‘𝐴) ∈ ℕ) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) = (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴)))
47	24, 22, 46	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) = (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴)))
48	45, 47	mpbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) = (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
49	48	oveq1d 7407	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))) = ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))))
50	43, 49	eqtrd 2796	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) = ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))))
51	50	oveq1d 7407	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) = (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
52	2, 11, 3	gexnnod 19611	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℕ)
53	15, 19, 28, 52	syl3anc 1389	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℕ)
54	53	nncnd 12223	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℂ)
55	24	nncnd 12223	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℂ)
56	24	nnne0d 12260	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ≠ 0)
57	54, 55, 56	divcan3d 11969	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) · (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴))) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) = (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)))
58	51, 57	eqtr2d 2797	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) = ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
59	2, 11, 3	gexnnod 19611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℕ)
60	15, 19, 41, 59	syl3anc 1389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℕ)
61	60	nncnd 12223	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℂ)
62	35	nncnd 12223	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℂ)
63	38	nncnd 12223	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℂ)
64	38	nnne0d 12260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≠ 0)
65	31	nncnd 12223	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℂ)
66	35	nnne0d 12260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≠ 0)
67	65, 62, 66	divcan1d 11965	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) = (𝑂‘𝑥))
68	2, 3, 26	odmulg 19579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ) → (𝑂‘𝑥) = ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
69	15, 29, 39, 68	syl3anc 1389	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) = ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
70	35	nnzd 12591	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℤ)
71		dvdsmul1 16294	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℤ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
72	39, 70, 71	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
73	72, 67	breqtrd 5125	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (𝑂‘𝑥))
74		gcdeq 16570	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ ∧ (𝑂‘𝑥) ∈ ℕ) → ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) = ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ↔ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (𝑂‘𝑥)))
75	38, 31, 74	syl2anc 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) = ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ↔ ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (𝑂‘𝑥)))
76	73, 75	mpbird 259	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) = ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
77	76	oveq1d 7407	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) gcd (𝑂‘𝑥)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
78	67, 69, 77	3eqtrrd 2801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
79	61, 62, 63, 64, 78	mulcanad 11819	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) = (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))
80	58, 79	oveq12d 7410	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) gcd (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) gcd (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
81		nndivdvds 16278	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑂‘𝐴) ∈ ℕ ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℕ))
82	22, 24, 81	syl2anc 593	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℕ))
83	45, 82	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℕ)
84	83	nnzd 12591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℤ)
85	84, 70	gcdcomd 16531	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) gcd (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) = ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))))
86		pcndvds2 16887	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝐴) ∈ ℕ) → ¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
87	18, 22, 86	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
88		coprm 16729	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℤ) → (¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ↔ (𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1))
89	18, 84, 88	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ↔ (𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1))
90	87, 89	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1)
91		prmz 16692	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
92	91	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
93		rpexp1i 16741	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ ℤ ∧ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℤ ∧ (𝑝 pCnt (𝑂‘𝑥)) ∈ ℕ₀) → ((𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1 → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1))
94	92, 84, 34, 93	syl3anc 1389	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1 → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1))
95	90, 94	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1)
96	80, 85, 95	3eqtrd 2800	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) gcd (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = 1)
97		eqid 2761	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
98	3, 2, 97	odadd 19873	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Abel ∧ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋 ∧ (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋) ∧ ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) gcd (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = 1) → (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
99	14, 28, 41, 96, 98	syl31anc 1391	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
100	58, 79	oveq12d 7410	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) · (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
101	99, 100	eqtrd 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) = (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
102		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑦 = (((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) → (𝑂‘𝑦) = (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))))
103	102	breq1d 5109	. . . . . . . . . . 11 ⊢ (𝑦 = (((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) → ((𝑂‘𝑦) ≤ (𝑂‘𝐴) ↔ (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) ≤ (𝑂‘𝐴)))
104		gexexlem.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑂‘𝑦) ≤ (𝑂‘𝐴))
105	104	ralrimiva 3153	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
106	105	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
107	2, 97	grpcl 18966	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋 ∧ (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ 𝑋)
108	15, 28, 41, 107	syl3anc 1389	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ 𝑋)
109	103, 106, 108	rspcdva 3582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)(+_g‘𝐺)(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥))) ≤ (𝑂‘𝐴))
110	101, 109	eqbrtrrd 5123	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≤ (𝑂‘𝐴))
111	83	nnred 12222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℝ)
112	22	nnred 12222	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℝ)
113	35	nnrpd 13032	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
114	111, 112, 113	lemuldivd 13083	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≤ (𝑂‘𝐴) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))))
115	110, 114	mpbid 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
116		nnrp 13002	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
117		nnrp 13002	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺)
118		nnrp 13002	. . . . . . . . . 10 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℝ⁺)
119		rpregt0 13005	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
120		rpregt0 13005	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
121		rpregt0 13005	. . . . . . . . . . 11 ⊢ ((𝑂‘𝐴) ∈ ℝ⁺ → ((𝑂‘𝐴) ∈ ℝ ∧ 0 < (𝑂‘𝐴)))
122		lediv2 12079	. . . . . . . . . . 11 ⊢ ((((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∧ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∧ ((𝑂‘𝐴) ∈ ℝ ∧ 0 < (𝑂‘𝐴))) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))))
123	119, 120, 121, 122	syl3an 1172	. . . . . . . . . 10 ⊢ (((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺ ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺ ∧ (𝑂‘𝐴) ∈ ℝ⁺) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))))
124	116, 117, 118, 123	syl3an 1172	. . . . . . . . 9 ⊢ (((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ ∧ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))))
125	35, 24, 22, 124	syl3anc 1389	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ↔ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))))
126	115, 125	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
127	17	nnred 12222	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
128	34	nn0zd 12590	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℤ)
129	23	nn0zd 12590	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℤ)
130		prmuz2 16713	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
131	130	adantl 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ_≥‘2))
132		eluz2gt1 12918	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘2) → 1 < 𝑝)
133	131, 132	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 1 < 𝑝)
134	127, 128, 129, 133	leexp2d 14262	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)) ↔ (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
135	126, 134	mpbird 259	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
136	135	ralrimiva 3153	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
137	2, 3	odcl 19559	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ₀)
138	137	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ₀)
139	138	nn0zd 12590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
140	5	nn0zd 12590	. . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℤ)
141	140	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
142		pc2dvds 16898	. . . . . 6 ⊢ (((𝑂‘𝑥) ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → ((𝑂‘𝑥) ∥ (𝑂‘𝐴) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴))))
143	139, 141, 142	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ (𝑂‘𝐴) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴))))
144	136, 143	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ (𝑂‘𝐴))
145	144	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴))
146	2, 11, 3	gexdvds2 19608	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑂‘𝐴) ∈ ℤ) → (𝐸 ∥ (𝑂‘𝐴) ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴)))
147	10, 140, 146	syl2anc 593	. . 3 ⊢ (𝜑 → (𝐸 ∥ (𝑂‘𝐴) ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴)))
148	145, 147	mpbird 259	. 2 ⊢ (𝜑 → 𝐸 ∥ (𝑂‘𝐴))
149		dvdseq 16331	. 2 ⊢ ((((𝑂‘𝐴) ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀) ∧ ((𝑂‘𝐴) ∥ 𝐸 ∧ 𝐸 ∥ (𝑂‘𝐴))) → (𝑂‘𝐴) = 𝐸)
150	5, 7, 13, 148, 149	syl22anc 849	1 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℝcr 11069 0cc0 11070 1c1 11071 · cmul 11075 < clt 11213 ≤ cle 11214 / cdiv 11841 ℕcn 12207 2c2 12269 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ℝ⁺crp 12990 ↑cexp 14071 ∥ cdvds 16269 gcd cgcd 16511 ℙcprime 16688 pCnt cpc 16855 Basecbs 17228 +_gcplusg 17269 Grpcgrp 18958 ._gcmg 19092 odcod 19547 gExcgex 19548 Abelcabl 19804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-q 12947 df-rp 12991 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-dvds 16270 df-gcd 16512 df-prm 16689 df-pc 16856 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-minusg 18962 df-sbg 18963 df-mulg 19093 df-od 19551 df-gex 19552 df-cmn 19805 df-abl 19806

This theorem is referenced by: gexex 19876

W3C validator