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Theorem gexexlem 19885

Description: Lemma for gexex 19886. (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 19569	. . 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 12585	. 2 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
8		gexexlem.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
9		ablgrp 19818	. . . 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 19619	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸)
13	10, 1, 12	syl2anc 584	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∥ 𝐸)
14	8	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Abel)
15	10	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Grp)
16		prmnn 16708	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
17	16	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
18		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
19	6	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐸 ∈ ℕ)
20	1	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ 𝑋)
21	2, 11, 3	gexnnod 19621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ)
22	15, 19, 20, 21	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℕ)
23	18, 22	pccld 16884	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℕ₀)
24	17, 23	nnexpcld 14281	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ)
25	24	nnzd 12638	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℤ)
26		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
27	2, 26	mulgcl 19122	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋)
29		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ 𝑋)
30	2, 11, 3	gexnnod 19621	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ)
31	15, 19, 29, 30	syl3anc 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℕ)
32		pcdvds 16898	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝑥) ∈ ℕ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥))
33	18, 31, 32	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∥ (𝑂‘𝑥))
34	18, 31	pccld 16884	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℕ₀)
35	17, 34	nnexpcld 14281	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ)
36		nndivdvds 16296	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ)
39	38	nnzd 12638	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ)
40	2, 26	mulgcl 19122	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋)
42	2, 3, 26	odmulg 19589	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 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 16898	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝐴) ∈ ℕ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴))
45	18, 22, 44	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∥ (𝑂‘𝐴))
46		gcdeq 16587	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) = (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
49	48	oveq1d 7446	. . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . 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 19621	. . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℕ)
54	53	nncnd 12280	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℂ)
55	24	nncnd 12280	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℂ)
56	24	nnne0d 12314	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ≠ 0)
57	54, 55, 56	divcan3d 12046	. . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 19621	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℕ)
61	60	nncnd 12280	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℂ)
62	35	nncnd 12280	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℂ)
63	38	nncnd 12280	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℂ)
64	38	nnne0d 12314	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≠ 0)
65	31	nncnd 12280	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℂ)
66	35	nnne0d 12314	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≠ 0)
67	65, 62, 66	divcan1d 12042	. . . . . . . . . . . . . . . 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 19589	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 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 12638	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℤ)
71		dvdsmul1 16312	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . 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 5174	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (𝑂‘𝑥))
74		gcdeq 16587	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 11896	. . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . 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 16296	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℕ)
84	83	nnzd 12638	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℤ)
85	84, 70	gcdcomd 16548	. . . . . . . . . . . . 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 16902	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ (𝑂‘𝐴) ∈ ℕ) → ¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
87	18, 22, 86	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ¬ 𝑝 ∥ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
88		coprm 16745	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1)
91		prmz 16709	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
92	91	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
93		rpexp1i 16757	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 2779	. . . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
98	3, 2, 97	odadd 19883	. . . . . . . . . . . 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 1372	. . . . . . . . . . 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 7449	. . . . . . . . . . 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 2775	. . . . . . . . . 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 6907	. . . . . . . . . . . 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 5158	. . . . . . . . . . 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 3144	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
106	105	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
107	2, 97	grpcl 18972	. . . . . . . . . . . 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 1370	. . . . . . . . . . 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 3623	. . . . . . . . . 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 5172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≤ (𝑂‘𝐴))
111	83	nnred 12279	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℝ)
112	22	nnred 12279	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℝ)
113	35	nnrpd 13073	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
114	111, 112, 113	lemuldivd 13124	. . . . . . . . 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 232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
116		nnrp 13044	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
117		nnrp 13044	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺)
118		nnrp 13044	. . . . . . . . . 10 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℝ⁺)
119		rpregt0 13047	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
120		rpregt0 13047	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
121		rpregt0 13047	. . . . . . . . . . 11 ⊢ ((𝑂‘𝐴) ∈ ℝ⁺ → ((𝑂‘𝐴) ∈ ℝ ∧ 0 < (𝑂‘𝐴)))
122		lediv2 12156	. . . . . . . . . . 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 1159	. . . . . . . . . 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 1159	. . . . . . . . 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 1370	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
127	17	nnred 12279	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
128	34	nn0zd 12637	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℤ)
129	23	nn0zd 12637	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℤ)
130		prmuz2 16730	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
131	130	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ_≥‘2))
132		eluz2gt1 12960	. . . . . . . . 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 14288	. . . . . . 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 257	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
136	135	ralrimiva 3144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
137	2, 3	odcl 19569	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ₀)
138	137	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ₀)
139	138	nn0zd 12637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
140	5	nn0zd 12637	. . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℤ)
141	140	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
142		pc2dvds 16913	. . . . . 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 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑂‘𝑥) ∥ (𝑂‘𝐴) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴))))
144	136, 143	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ (𝑂‘𝐴))
145	144	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴))
146	2, 11, 3	gexdvds2 19618	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑂‘𝐴) ∈ ℤ) → (𝐸 ∥ (𝑂‘𝐴) ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴)))
147	10, 140, 146	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐸 ∥ (𝑂‘𝐴) ↔ ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴)))
148	145, 147	mpbird 257	. 2 ⊢ (𝜑 → 𝐸 ∥ (𝑂‘𝐴))
149		dvdseq 16348	. 2 ⊢ ((((𝑂‘𝐴) ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀) ∧ ((𝑂‘𝐴) ∥ 𝐸 ∧ 𝐸 ∥ (𝑂‘𝐴))) → (𝑂‘𝐴) = 𝐸)
150	5, 7, 13, 148, 149	syl22anc 839	1 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ₀cn0 12524 ℤcz 12611 ℤ_≥cuz 12876 ℝ⁺crp 13032 ↑cexp 14099 ∥ cdvds 16287 gcd cgcd 16528 ℙcprime 16705 pCnt cpc 16870 Basecbs 17245 +_gcplusg 17298 Grpcgrp 18964 ._gcmg 19098 odcod 19557 gExcgex 19558 Abelcabl 19814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-od 19561 df-gex 19562 df-cmn 19815 df-abl 19816

This theorem is referenced by: gexex 19886

W3C validator