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Theorem mumullem2 27516

Description: Lemma for mumul 27517. 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 3122	. . . 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 490	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
3		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
4	2, 3	pccld 17037	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
5	4	nn0red 12675	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ)
6		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℕ)
7	2, 6	pccld 17037	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℕ₀)
8	7	nn0red 12675	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℝ)
9		1red 11316	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℝ)
10		le2add 11805	. . . . . . . 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 852	. . . . . . 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 11276	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
13		simpl3 1212	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) = 1)
14	13	oveq2d 7440	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 gcd 𝐵)) = (𝑝 pCnt 1))
15	3	nnzd 12726	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
16	6	nnzd 12726	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
17		pcgcd 17065	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 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 17042	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ ℙ → (𝑝 pCnt 1) = 0)
20	19	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 1) = 0)
21	14, 18, 20	3eqtr3d 2803	. . . . . . . . . . . . . 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 4522	. . . . . . . . . . . . . . . 16 ⊢ if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), 1, 1) = 1
23		ifeq12 4500	. . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 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 250	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)) → 1 = 0))
27	26	necon3ad 2968	. . . . . . . . . . . 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 21	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ¬ (1 = (𝑝 pCnt 𝐴) ∧ 1 = (𝑝 pCnt 𝐵)))
29		ax-1cn 11265	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
30	5	recnd 11344	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℂ)
31		subeq0 11591	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐴) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
32	29, 30, 31	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
33	8	recnd 11344	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℂ)
34		subeq0 11591	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐵) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
35	29, 33, 34	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
36	32, 35	anbi12d 644	. . . . . . . . . . 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 328	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ¬ ((1 − (𝑝 pCnt 𝐴)) = 0 ∧ (1 − (𝑝 pCnt 𝐵)) = 0))
38	37	adantr 486	. . . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ ((1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1))
40		1re 11315	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
41	40, 40	readdcli 11331	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
42	41	recni 11330	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) ∈ ℂ
43	4, 7	nn0addcld 12678	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℕ₀)
44	43	nn0red 12675	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℝ)
45	44	recnd 11344	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℂ)
46		subeq0 11591	. . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . 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 11344	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
50	49, 49, 30, 33	addsub4d 11723	. . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 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 284	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 11836	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
55	40, 5, 54	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
56		subge0 11836	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
57	40, 8, 56	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
58	55, 57	anbi12d 644	. . . . . . . . . . . . . 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 11629	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
60	40, 5, 59	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
61		resubcl 11629	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
62	40, 8, 61	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
63		add20 11835	. . . . . . . . . . . . . . . . 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 673	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 263	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 286	. . . . . . . . . 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 2991	. . . . . . . . 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 260	. . . . . . . 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 418	. . . . . . 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 522	. . . . . 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 12721	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
76		nnne0 12379	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
77	75, 76	jca 521	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0))
78	3, 77	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐴 ∈ ℤ ∧ 𝐴 ≠ 0))
79		nnz 12721	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
80		nnne0 12379	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0)
81	79, 80	jca 521	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
82	6, 81	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0))
83		pcmul 17038	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 · 𝐵)) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))
85	84	breq1d 5112	. . . . . . 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 12629	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
87		nn0leltp1 12765	. . . . . . . 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 598	. . . . . . 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 11418	. . . . . . . 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 598	. . . . . . 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 308	. . . . . 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 262	. . . . 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 3174	. . . 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 246	. . 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 27472	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))
96		issqf 27472	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((μ‘𝐵) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1))
97	95, 96	bi2anan9 650	. . . 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 1150	. . 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 12366	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
100	99	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 · 𝐵) ∈ ℕ)
101		issqf 27472	. . . 4 ⊢ ((𝐴 · 𝐵) ∈ ℕ → ((μ‘(𝐴 · 𝐵)) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt (𝐴 · 𝐵)) ≤ 1))
102	100, 101	syl 18	. . 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 297	. 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (((μ‘𝐴) ≠ 0 ∧ (μ‘𝐵) ≠ 0) → (μ‘(𝐴 · 𝐵)) ≠ 0))
104	103	imp 412	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ifcif 4481 class class class wbr 5102 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 ≤ cle 11351 − cmin 11548 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 gcd cgcd 16673 ℙcprime 16855 pCnt cpc 17023 μcmu 27431

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9441 df-inf 9442 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-n0 12614 df-z 12701 df-uz 12973 df-q 13083 df-rp 13128 df-fz 13647 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-dvds 16432 df-gcd 16674 df-prm 16856 df-pc 17024 df-mu 27437

This theorem is used by: mumul 27517

W3C validator