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Theorem gexexlem 19630

Description: Lemma for gexex 19631. (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 19318	. . 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 12473	. 2 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
8		gexexlem.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Abel)
9		ablgrp 19567	. . . 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 19368	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∥ 𝐸)
13	10, 1, 12	syl2anc 584	. 2 ⊢ (𝜑 → (𝑂‘𝐴) ∥ 𝐸)
14	8	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Abel)
15	10	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐺 ∈ Grp)
16		prmnn 16550	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ)
17	16	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℕ)
18		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
19	6	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐸 ∈ ℕ)
20	1	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ 𝑋)
21	2, 11, 3	gexnnod 19370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝐴 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℕ)
22	15, 19, 20, 21	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℕ)
23	18, 22	pccld 16722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℕ₀)
24	17, 23	nnexpcld 14148	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ)
25	24	nnzd 12526	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℤ)
26		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (._g‘𝐺) = (._g‘𝐺)
27	2, 26	mulgcl 18893	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴) ∈ 𝑋)
29		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑥 ∈ 𝑋)
30	2, 11, 3	gexnnod 19370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ Grp ∧ 𝐸 ∈ ℕ ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ)
31	15, 19, 29, 30	syl3anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℕ)
32		pcdvds 16736	. . . . . . . . . . . . . . . 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 16722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℕ₀)
35	17, 34	nnexpcld 14148	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ)
36		nndivdvds 16145	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℕ)
39	38	nnzd 12526	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℤ)
40	2, 26	mulgcl 18893	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥) ∈ 𝑋)
42	2, 3, 26	odmulg 19338	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 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 16736	. . . . . . . . . . . . . . . . . . . 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 16434	. . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) gcd (𝑂‘𝐴)) = (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
49	48	oveq1d 7372	. . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . 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 19370	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℕ)
54	53	nncnd 12169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘((𝑝↑(𝑝 pCnt (𝑂‘𝐴)))(._g‘𝐺)𝐴)) ∈ ℂ)
55	24	nncnd 12169	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℂ)
56	24	nnne0d 12203	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ≠ 0)
57	54, 55, 56	divcan3d 11936	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 19370	. . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℕ)
61	60	nncnd 12169	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘(((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥))))(._g‘𝐺)𝑥)) ∈ ℂ)
62	35	nncnd 12169	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℂ)
63	38	nncnd 12169	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∈ ℂ)
64	38	nnne0d 12203	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≠ 0)
65	31	nncnd 12169	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝑥) ∈ ℂ)
66	35	nnne0d 12203	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≠ 0)
67	65, 62, 66	divcan1d 11932	. . . . . . . . . . . . . . . 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 19338	. . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . 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 12526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℤ)
71		dvdsmul1 16160	. . . . . . . . . . . . . . . . . . . 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 5131	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝑥) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ∥ (𝑂‘𝑥))
74		gcdeq 16434	. . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . 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 11790	. . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . 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 16145	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℕ)
84	83	nnzd 12526	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℤ)
85	84, 70	gcdcomd 16394	. . . . . . . . . . . . 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 16740	. . . . . . . . . . . . . . . 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 16587	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 gcd ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))) = 1)
91		prmz 16551	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
92	91	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℤ)
93		rpexp1i 16599	. . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
98	3, 2, 97	odadd 19628	. . . . . . . . . . . 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 1373	. . . . . . . . . . 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 7375	. . . . . . . . . . 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 2776	. . . . . . . . . 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 6842	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 3143	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
106	105	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ∀𝑦 ∈ 𝑋 (𝑂‘𝑦) ≤ (𝑂‘𝐴))
107	2, 97	grpcl 18756	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 5129	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) · (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))) ≤ (𝑂‘𝐴))
111	83	nnred 12168	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ∈ ℝ)
112	22	nnred 12168	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑂‘𝐴) ∈ ℝ)
113	35	nnrpd 12955	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
114	111, 112, 113	lemuldivd 13006	. . . . . . . . 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 231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))) ≤ ((𝑂‘𝐴) / (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
116		nnrp 12926	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺)
117		nnrp 12926	. . . . . . . . . 10 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℕ → (𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺)
118		nnrp 12926	. . . . . . . . . 10 ⊢ ((𝑂‘𝐴) ∈ ℕ → (𝑂‘𝐴) ∈ ℝ⁺)
119		rpregt0 12929	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝑥)))))
120		rpregt0 12929	. . . . . . . . . . 11 ⊢ ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ⁺ → ((𝑝↑(𝑝 pCnt (𝑂‘𝐴))) ∈ ℝ ∧ 0 < (𝑝↑(𝑝 pCnt (𝑂‘𝐴)))))
121		rpregt0 12929	. . . . . . . . . . 11 ⊢ ((𝑂‘𝐴) ∈ ℝ⁺ → ((𝑂‘𝐴) ∈ ℝ ∧ 0 < (𝑂‘𝐴)))
122		lediv2 12045	. . . . . . . . . . 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 1160	. . . . . . . . . 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 1160	. . . . . . . . 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 1371	. . . . . . . 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 256	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝↑(𝑝 pCnt (𝑂‘𝑥))) ≤ (𝑝↑(𝑝 pCnt (𝑂‘𝐴))))
127	17	nnred 12168	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℝ)
128	34	nn0zd 12525	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ∈ ℤ)
129	23	nn0zd 12525	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝐴)) ∈ ℤ)
130		prmuz2 16572	. . . . . . . . . 10 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ (ℤ_≥‘2))
131	130	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ (ℤ_≥‘2))
132		eluz2gt1 12845	. . . . . . . . 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 14155	. . . . . . 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 256	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
136	135	ralrimiva 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑝 ∈ ℙ (𝑝 pCnt (𝑂‘𝑥)) ≤ (𝑝 pCnt (𝑂‘𝐴)))
137	2, 3	odcl 19318	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → (𝑂‘𝑥) ∈ ℕ₀)
138	137	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℕ₀)
139	138	nn0zd 12525	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∈ ℤ)
140	5	nn0zd 12525	. . . . . . 7 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℤ)
141	140	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝐴) ∈ ℤ)
142		pc2dvds 16751	. . . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑂‘𝑥) ∥ (𝑂‘𝐴))
145	144	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑂‘𝑥) ∥ (𝑂‘𝐴))
146	2, 11, 3	gexdvds2 19367	. . . 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 256	. 2 ⊢ (𝜑 → 𝐸 ∥ (𝑂‘𝐴))
149		dvdseq 16196	. 2 ⊢ ((((𝑂‘𝐴) ∈ ℕ₀ ∧ 𝐸 ∈ ℕ₀) ∧ ((𝑂‘𝐴) ∥ 𝐸 ∧ 𝐸 ∥ (𝑂‘𝐴))) → (𝑂‘𝐴) = 𝐸)
150	5, 7, 13, 148, 149	syl22anc 837	1 ⊢ (𝜑 → (𝑂‘𝐴) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 < clt 11189 ≤ cle 11190 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ℝ⁺crp 12915 ↑cexp 13967 ∥ cdvds 16136 gcd cgcd 16374 ℙcprime 16547 pCnt cpc 16708 Basecbs 17083 +_gcplusg 17133 Grpcgrp 18748 ._gcmg 18872 odcod 19306 gExcgex 19307 Abelcabl 19563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-od 19310 df-gex 19311 df-cmn 19564 df-abl 19565

This theorem is referenced by: gexex 19631

W3C validator