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Theorem pcgcd1 16806

Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

**Assertion**
Ref	Expression
pcgcd1	⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))

**Proof of Theorem pcgcd1**
Step	Hyp	Ref	Expression
1		oveq2 7413	. . . 4 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
2	1	oveq2d 7421	. . 3 ⊢ (𝐵 = 0 → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt (𝐴 gcd 0)))
3	2	eqeq1d 2734	. 2 ⊢ (𝐵 = 0 → ((𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴) ↔ (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴)))
4		simpl1 1191	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝑃 ∈ ℙ)
5		simp2 1137	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
6	5	adantr 481	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℤ)
7		simpl3 1193	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℤ)
8		simprr 771	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
9		simpr 485	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → 𝐵 = 0)
10	9	necon3ai 2965	. . . . . . . 8 ⊢ (𝐵 ≠ 0 → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
11	8, 10	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ¬ (𝐴 = 0 ∧ 𝐵 = 0))
12		gcdn0cl 16439	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
13	6, 7, 11, 12	syl21anc 836	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	13	nnzd 12581	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
15		gcddvds 16440	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
16	6, 7, 15	syl2anc 584	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
17	16	simpld 495	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∥ 𝐴)
18		pcdvdstr 16805	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴 gcd 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝐵) ∥ 𝐴)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ≤ (𝑃 pCnt 𝐴))
19	4, 14, 6, 17, 18	syl13anc 1372	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ≤ (𝑃 pCnt 𝐴))
20		zq 12934	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ)
21	6, 20	syl 17	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℚ)
22		pcxcl 16790	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
23	4, 21, 22	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
24		pczcl 16777	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
25	4, 7, 8, 24	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
26	25	nn0red 12529	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℝ)
27		pcge0 16791	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → 0 ≤ (𝑃 pCnt 𝐴))
28	4, 6, 27	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 0 ≤ (𝑃 pCnt 𝐴))
29		ge0gtmnf 13147	. . . . . . . . . 10 ⊢ (((𝑃 pCnt 𝐴) ∈ ℝ^* ∧ 0 ≤ (𝑃 pCnt 𝐴)) → -∞ < (𝑃 pCnt 𝐴))
30	23, 28, 29	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → -∞ < (𝑃 pCnt 𝐴))
31		simprl 769	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
32		xrre 13144	. . . . . . . . 9 ⊢ ((((𝑃 pCnt 𝐴) ∈ ℝ^* ∧ (𝑃 pCnt 𝐵) ∈ ℝ) ∧ (-∞ < (𝑃 pCnt 𝐴) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))) → (𝑃 pCnt 𝐴) ∈ ℝ)
33	23, 26, 30, 31, 32	syl22anc 837	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℝ)
34		pnfnre 11251	. . . . . . . . . . . 12 ⊢ +∞ ∉ ℝ
35	34	neli 3048	. . . . . . . . . . 11 ⊢ ¬ +∞ ∈ ℝ
36		pc0 16783	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
37	4, 36	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 0) = +∞)
38	37	eleq1d 2818	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt 0) ∈ ℝ ↔ +∞ ∈ ℝ))
39	35, 38	mtbiri 326	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ¬ (𝑃 pCnt 0) ∈ ℝ)
40		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝐴 = 0 → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0))
41	40	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → ((𝑃 pCnt 𝐴) ∈ ℝ ↔ (𝑃 pCnt 0) ∈ ℝ))
42	41	notbid 317	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (¬ (𝑃 pCnt 𝐴) ∈ ℝ ↔ ¬ (𝑃 pCnt 0) ∈ ℝ))
43	39, 42	syl5ibrcom 246	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 = 0 → ¬ (𝑃 pCnt 𝐴) ∈ ℝ))
44	43	necon2ad 2955	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt 𝐴) ∈ ℝ → 𝐴 ≠ 0))
45	33, 44	mpd 15	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐴 ≠ 0)
46		pczdvds 16792	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
47	4, 6, 45, 46	syl12anc 835	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
48		pczcl 16777	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
49	4, 6, 45, 48	syl12anc 835	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
50		pcdvdsb 16798	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ (𝑃 pCnt 𝐴) ∈ ℕ₀) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵))
51	4, 7, 49, 50	syl3anc 1371	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵))
52	31, 51	mpbid 231	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)
53		prmnn 16607	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
54	4, 53	syl 17	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝑃 ∈ ℕ)
55	54, 49	nnexpcld 14204	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
56	55	nnzd 12581	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
57		dvdsgcd 16482	. . . . . . 7 ⊢ (((𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵)))
58	56, 6, 7, 57	syl3anc 1371	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (((𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴 ∧ (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵)))
59	47, 52, 58	mp2and 697	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵))
60		pcdvdsb 16798	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 gcd 𝐵) ∈ ℤ ∧ (𝑃 pCnt 𝐴) ∈ ℕ₀) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 gcd 𝐵)) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵)))
61	4, 14, 49, 60	syl3anc 1371	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 gcd 𝐵)) ↔ (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵)))
62	59, 61	mpbird 256	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 gcd 𝐵)))
63	4, 13	pccld 16779	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ∈ ℕ₀)
64	63	nn0red 12529	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ∈ ℝ)
65	64, 33	letri3d 11352	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴) ↔ ((𝑃 pCnt (𝐴 gcd 𝐵)) ≤ (𝑃 pCnt 𝐴) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt (𝐴 gcd 𝐵)))))
66	19, 62, 65	mpbir2and 711	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))
67	66	anassrs 468	. 2 ⊢ ((((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) ∧ 𝐵 ≠ 0) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))
68		gcdid0 16457	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 gcd 0) = (abs‘𝐴))
69	5, 68	syl 17	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 0) = (abs‘𝐴))
70	69	oveq2d 7421	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt (abs‘𝐴)))
71		pcabs 16804	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
72	20, 71	sylan2 593	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
73	72	3adant3 1132	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
74	70, 73	eqtrd 2772	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴))
75	74	adantr 481	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴))
76	3, 67, 75	pm2.61ne 3027	1 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℚcq 12928 ↑cexp 14023 abscabs 15177 ∥ cdvds 16193 gcd cgcd 16431 ℙcprime 16604 pCnt cpc 16765

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766

This theorem is referenced by: pcgcd 16807

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