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Theorem mumullem2 27241

Description: Lemma for mumul 27242. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

**Assertion**
Ref	Expression
mumullem2	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) ≠ 0)

Proof of Theorem mumullem2 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3117	. . . 4 ⊢ (∀𝑝 ∈ ℙ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1 ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1))
2		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
3		simpl1 1191	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
4	2, 3	pccld 16897	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
5	4	nn0red 12614	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ)
6		simpl2 1192	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℕ)
7	2, 6	pccld 16897	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℕ₀)
8	7	nn0red 12614	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℝ)
9		1red 11291	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℝ)
10		le2add 11772	. . . . . . . 8 ⊢ ((((𝑝 pCnt 𝐴) ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) ∧ (1 ∈ ℝ ∧ 1 ∈ ℝ)) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1)))
11	5, 8, 9, 9, 10	syl22anc 838	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1)))
12		ax-1ne0 11253	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
13		simpl3 1193	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) = 1)
14	13	oveq2d 7464	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 gcd 𝐵)) = (𝑝 pCnt 1))
15	3	nnzd 12666	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
16	6	nnzd 12666	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
17		pcgcd 16925	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑝 pCnt (𝐴 gcd 𝐵)) = if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)))
18	2, 15, 16, 17	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 gcd 𝐵)) = if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)))
19		pc1 16902	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → (𝑝 pCnt 1) = 0)
20	19	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 1) = 0)
21	14, 18, 20	3eqtr3d 2788	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)) = 0)
22		ifid 4588	. . . . . . . . . . . . . . . 16 ⊢ if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), 1, 1) = 1
23		ifeq12 4566	. . . . . . . . . . . . . . . 16 ⊢ ((1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)) → if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), 1, 1) = if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)))
24	22, 23	eqtr3id 2794	. . . . . . . . . . . . . . 15 ⊢ ((1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)) → 1 = if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)))
25	24	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ ((1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)) → (1 = 0 ↔ if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), (𝑝 pCnt 𝐴), (𝑝 pCnt 𝐵)) = 0))
26	21, 25	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)) → 1 = 0))
27	26	necon3ad 2959	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 ≠ 0 → ¬ (1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵))))
28	12, 27	mpi 20	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ¬ (1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)))
29		ax-1cn 11242	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
30	5	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℂ)
31		subeq0 11562	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐴) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
32	29, 30, 31	sylancr 586	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
33	8	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℂ)
34		subeq0 11562	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐵) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
35	29, 33, 34	sylancr 586	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
36	32, 35	anbi12d 631	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0) ↔ (1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵))))
37	28, 36	mtbird 325	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ¬ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))
38	37	adantr 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → ¬ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))
39		eqcom 2747	. . . . . . . . . . 11 ⊢ ((1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1))
40		1re 11290	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
41	40, 40	readdcli 11305	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
42	41	recni 11304	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) ∈ ℂ
43	4, 7	nn0addcld 12617	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℕ₀)
44	43	nn0red 12614	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℝ)
45	44	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℂ)
46		subeq0 11562	. . . . . . . . . . . . . . . 16 ⊢ (((1 + 1) ∈ ℂ ∧ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℂ) → (((1 + 1) − ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))) = 0 ↔ (1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))))
47	42, 45, 46	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((1 + 1) − ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))) = 0 ↔ (1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))))
48	47, 39	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((1 + 1) − ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))) = 0 ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1)))
49	9	recnd 11318	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
50	49, 49, 30, 33	addsub4d 11694	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 + 1) − ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))) = ((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))))
51	50	eqeq1d 2742	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((1 + 1) − ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0))
52	48, 51	bitr3d 281	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1) ↔ ((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0))
53	52	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1) ↔ ((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0))
54		subge0 11803	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
55	40, 5, 54	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
56		subge0 11803	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
57	40, 8, 56	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
58	55, 57	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((0 ≤ (1 − (𝑝 pCnt 𝐴)) ∧ 0 ≤ (1 − (𝑝 pCnt 𝐵))) ↔ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)))
59		resubcl 11600	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
60	40, 5, 59	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
61		resubcl 11600	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
62	40, 8, 61	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
63		add20 11802	. . . . . . . . . . . . . . . . 17 ⊢ ((((1 − (𝑝 pCnt 𝐴)) ∈ ℝ ∧ 0 ≤ (1 − (𝑝 pCnt 𝐴))) ∧ ((1 − (𝑝 pCnt 𝐵)) ∈ ℝ ∧ 0 ≤ (1 − (𝑝 pCnt 𝐵)))) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
64	63	an4s 659	. . . . . . . . . . . . . . . 16 ⊢ ((((1 − (𝑝 pCnt 𝐴)) ∈ ℝ ∧ (1 − (𝑝 pCnt 𝐵)) ∈ ℝ) ∧ (0 ≤ (1 − (𝑝 pCnt 𝐴)) ∧ 0 ≤ (1 − (𝑝 pCnt 𝐵)))) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
65	64	ex 412	. . . . . . . . . . . . . . 15 ⊢ (((1 − (𝑝 pCnt 𝐴)) ∈ ℝ ∧ (1 − (𝑝 pCnt 𝐵)) ∈ ℝ) → ((0 ≤ (1 − (𝑝 pCnt 𝐴)) ∧ 0 ≤ (1 − (𝑝 pCnt 𝐵))) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))))
66	60, 62, 65	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((0 ≤ (1 − (𝑝 pCnt 𝐴)) ∧ 0 ≤ (1 − (𝑝 pCnt 𝐵))) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))))
67	58, 66	sylbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))))
68	67	imp 406	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → (((1 − (𝑝 pCnt 𝐴)) + (1 − (𝑝 pCnt 𝐵))) = 0 ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
69	53, 68	bitrd 279	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1) ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
70	39, 69	bitrid 283	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → ((1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ↔ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
71	70	necon3abid 2983	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → ((1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ↔ ¬ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0)))
72	38, 71	mpbird 257	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) ∧ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1)) → (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))
73	72	ex 412	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵))))
74	11, 73	jcad 512	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1) ∧ (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))))
75		nnz 12660	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
76		nnne0 12327	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
77	75, 76	jca 511	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0))
78	3, 77	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0))
79		nnz 12660	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
80		nnne0 12327	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
81	79, 80	jca 511	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
82	6, 81	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
83		pcmul 16898	. . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0)) → (𝑝 pCnt (𝐴 · 𝐵)) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))
84	2, 78, 82, 83	syl3anc 1371	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 · 𝐵)) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))
85	84	breq1d 5176	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt (𝐴 · 𝐵)) ≤ 1 ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ 1))
86		1nn0 12569	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
87		nn0leltp1 12702	. . . . . . . 8 ⊢ ((((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ 1 ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) < (1 + 1)))
88	43, 86, 87	sylancl 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ 1 ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) < (1 + 1)))
89		ltlen 11391	. . . . . . . 8 ⊢ ((((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℝ ∧ (1 + 1) ∈ ℝ) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) < (1 + 1) ↔ (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1) ∧ (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))))
90	44, 41, 89	sylancl 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) < (1 + 1) ↔ (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1) ∧ (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))))
91	85, 88, 90	3bitrd 305	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt (𝐴 · 𝐵)) ≤ 1 ↔ (((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ≤ (1 + 1) ∧ (1 + 1) ≠ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))))
92	74, 91	sylibrd 259	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
93	92	ralimdva 3173	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (∀𝑝 ∈ ℙ ((𝑝 pCnt 𝐴) ≤ 1 ∧ (𝑝 pCnt 𝐵) ≤ 1) → ∀𝑝 ∈ ℙ (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
94	1, 93	biimtrrid 243	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1 ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1) → ∀𝑝 ∈ ℙ (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
95		issqf 27197	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))
96		issqf 27197	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((μ‘𝐵) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1))
97	95, 96	bi2anan9 637	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1 ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1)))
98	97	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0) ↔ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1 ∧ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1)))
99		nnmulcl 12317	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
100	99	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 · 𝐵) ∈ ℕ)
101		issqf 27197	. . . 4 ⊢ ((𝐴 · 𝐵) ∈ ℕ → ((μ‘(𝐴 · 𝐵)) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
102	100, 101	syl 17	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → ((μ‘(𝐴 · 𝐵)) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
103	94, 98, 102	3imtr4d 294	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0) → (μ‘(𝐴 · 𝐵)) ≠ 0))
104	103	imp 406	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ ((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0)) → (μ‘(𝐴 · 𝐵)) ≠ 0)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ifcif 4548 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 gcd cgcd 16540 ℙcprime 16718 pCnt cpc 16883 μcmu 27156

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-fz 13568 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-gcd 16541 df-prm 16719 df-pc 16884 df-mu 27162

This theorem is referenced by: mumul 27242

W3C validator