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Theorem pcgcd1 16805

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 7366	. . . 4 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
2	1	oveq2d 7374	. . 3 ⊢ (𝐵 = 0 → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt (𝐴 gcd 0)))
3	2	eqeq1d 2738	. 2 ⊢ (𝐵 = 0 → ((𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴) ↔ (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴)))
4		simpl1 1192	. . . . 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 480	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐴 ∈ ℤ)
7		simpl3 1194	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℤ)
8		simprr 772	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐵 ≠ 0)
9		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ 𝐵 = 0) → 𝐵 = 0)
10	9	necon3ai 2957	. . . . . . . 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 16429	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
13	6, 7, 11, 12	syl21anc 837	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	13	nnzd 12514	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
15		gcddvds 16430	. . . . . . 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 494	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∥ 𝐴)
18		pcdvdstr 16804	. . . . 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 1374	. . . 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 12867	. . . . . . . . . . 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 16789	. . . . . . . . . 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 16776	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
25	4, 7, 8, 24	syl12anc 836	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
26	25	nn0red 12463	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℝ)
27		pcge0 16790	. . . . . . . . . . 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 13087	. . . . . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
32		xrre 13084	. . . . . . . . 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 838	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℝ)
34		pnfnre 11173	. . . . . . . . . . . 12 ⊢ +∞ ∉ ℝ
35	34	neli 3038	. . . . . . . . . . 11 ⊢ ¬ +∞ ∈ ℝ
36		pc0 16782	. . . . . . . . . . . . 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 2821	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ((𝑃 pCnt 0) ∈ ℝ ↔ +∞ ∈ ℝ))
39	35, 38	mtbiri 327	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → ¬ (𝑃 pCnt 0) ∈ ℝ)
40		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝐴 = 0 → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0))
41	40	eleq1d 2821	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → ((𝑃 pCnt 𝐴) ∈ ℝ ↔ (𝑃 pCnt 0) ∈ ℝ))
42	41	notbid 318	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (¬ (𝑃 pCnt 𝐴) ∈ ℝ ↔ ¬ (𝑃 pCnt 0) ∈ ℝ))
43	39, 42	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 = 0 → ¬ (𝑃 pCnt 𝐴) ∈ ℝ))
44	43	necon2ad 2947	. . . . . . . 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 16791	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
47	4, 6, 45, 46	syl12anc 836	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
48		pczcl 16776	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
49	4, 6, 45, 48	syl12anc 836	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
50		pcdvdsb 16797	. . . . . . . 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 1373	. . . . . . 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 232	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐵)
53		prmnn 16601	. . . . . . . . . 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 14168	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
56	55	nnzd 12514	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
57		dvdsgcd 16471	. . . . . . 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 1373	. . . . . 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 699	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵))
60		pcdvdsb 16797	. . . . . 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 1373	. . . . 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 257	. . . 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 16778	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ∈ ℕ₀)
64	63	nn0red 12463	. . . . 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 11275	. . . 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 713	. . 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 467	. 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 16447	. . . . . 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 7374	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt (abs‘𝐴)))
71		pcabs 16803	. . . . . 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 2771	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴))
75	74	adantr 480	. 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 3017	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 0cc0 11026 +∞cpnf 11163 -∞cmnf 11164 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 ℕcn 12145 ℕ₀cn0 12401 ℤcz 12488 ℚcq 12861 ↑cexp 13984 abscabs 15157 ∥ cdvds 16179 gcd cgcd 16421 ℙcprime 16598 pCnt cpc 16764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765

This theorem is referenced by: pcgcd 16806

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