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Theorem pcgcd1 16506

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 7263	. . . 4 ⊢ (𝐵 = 0 → (𝐴 gcd 𝐵) = (𝐴 gcd 0))
2	1	oveq2d 7271	. . 3 ⊢ (𝐵 = 0 → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt (𝐴 gcd 0)))
3	2	eqeq1d 2740	. 2 ⊢ (𝐵 = 0 → ((𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴) ↔ (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt 𝐴)))
4		simpl1 1189	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝑃 ∈ ℙ)
5		simp2 1135	. . . . . . . 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 1191	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 𝐵 ∈ ℤ)
8		simprr 769	. . . . . . . 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 2967	. . . . . . . 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 16137	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
13	6, 7, 11, 12	syl21anc 834	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℕ)
14	13	nnzd 12354	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝐴 gcd 𝐵) ∈ ℤ)
15		gcddvds 16138	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵))
16	6, 7, 15	syl2anc 583	. . . . . 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 16505	. . . . 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 1370	. . . 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 12623	. . . . . . . . . . 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 16490	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
23	4, 21, 22	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℝ^*)
24		pczcl 16477	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
25	4, 7, 8, 24	syl12anc 833	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℕ₀)
26	25	nn0red 12224	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐵) ∈ ℝ)
27		pcge0 16491	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → 0 ≤ (𝑃 pCnt 𝐴))
28	4, 6, 27	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → 0 ≤ (𝑃 pCnt 𝐴))
29		ge0gtmnf 12835	. . . . . . . . . 10 ⊢ (((𝑃 pCnt 𝐴) ∈ ℝ^* ∧ 0 ≤ (𝑃 pCnt 𝐴)) → -∞ < (𝑃 pCnt 𝐴))
30	23, 28, 29	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → -∞ < (𝑃 pCnt 𝐴))
31		simprl 767	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
32		xrre 12832	. . . . . . . . 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 835	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℝ)
34		pnfnre 10947	. . . . . . . . . . . 12 ⊢ +∞ ∉ ℝ
35	34	neli 3050	. . . . . . . . . . 11 ⊢ ¬ +∞ ∈ ℝ
36		pc0 16483	. . . . . . . . . . . . 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 2823	. . . . . . . . . . 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 7263	. . . . . . . . . . . 12 ⊢ (𝐴 = 0 → (𝑃 pCnt 𝐴) = (𝑃 pCnt 0))
41	40	eleq1d 2823	. . . . . . . . . . 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 2957	. . . . . . . 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 16492	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
47	4, 6, 45, 46	syl12anc 833	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ 𝐴)
48		pczcl 16477	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
49	4, 6, 45, 48	syl12anc 833	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℕ₀)
50		pcdvdsb 16498	. . . . . . . 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 1369	. . . . . . 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 16307	. . . . . . . . . 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 13888	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℕ)
56	55	nnzd 12354	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∈ ℤ)
57		dvdsgcd 16180	. . . . . . 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 1369	. . . . . 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 695	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝐴)) ∥ (𝐴 gcd 𝐵))
60		pcdvdsb 16498	. . . . . 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 1369	. . . . 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 16479	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∧ 𝐵 ≠ 0)) → (𝑃 pCnt (𝐴 gcd 𝐵)) ∈ ℕ₀)
64	63	nn0red 12224	. . . . 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 11047	. . . 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 709	. . 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 16155	. . . . . 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 7271	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 0)) = (𝑃 pCnt (abs‘𝐴)))
71		pcabs 16504	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
72	20, 71	sylan2 592	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
73	72	3adant3 1130	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (abs‘𝐴)) = (𝑃 pCnt 𝐴))
74	70, 73	eqtrd 2778	. . 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 3029	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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℚcq 12617 ↑cexp 13710 abscabs 14873 ∥ cdvds 15891 gcd cgcd 16129 ℙcprime 16304 pCnt cpc 16465

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466

This theorem is referenced by: pcgcd 16507

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