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Theorem mumullem2 27223

Description: Lemma for mumul 27224. 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 3111	. . . 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 1192	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
4	2, 3	pccld 16888	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
5	4	nn0red 12588	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ)
6		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℕ)
7	2, 6	pccld 16888	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℕ₀)
8	7	nn0red 12588	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℝ)
9		1red 11262	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℝ)
10		le2add 11745	. . . . . . . 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 839	. . . . . . 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 11224	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
13		simpl3 1194	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝐴 gcd 𝐵) = 1)
14	13	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 gcd 𝐵)) = (𝑝 pCnt 1))
15	3	nnzd 12640	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℤ)
16	6	nnzd 12640	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℤ)
17		pcgcd 16916	. . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . 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 16893	. . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . 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 4566	. . . . . . . . . . . . . . . 16 ⊢ if((𝑝 pCnt 𝐴) ≤ (𝑝 pCnt 𝐵), 1, 1) = 1
23		ifeq12 4544	. . . . . . . . . . . . . . . 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 2791	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 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 2953	. . . . . . . . . . . 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 11213	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
30	5	recnd 11289	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℂ)
31		subeq0 11535	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐴) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
32	29, 30, 31	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐴)) = 0 ↔ 1 = (𝑝 pCnt 𝐴)))
33	8	recnd 11289	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℂ)
34		subeq0 11535	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ (𝑝 pCnt 𝐵) ∈ ℂ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
35	29, 33, 34	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((1 − (𝑝 pCnt 𝐵)) = 0 ↔ 1 = (𝑝 pCnt 𝐵)))
36	32, 35	anbi12d 632	. . . . . . . . . . 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 2744	. . . . . . . . . . 11 ⊢ ((1 + 1) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ↔ ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) = (1 + 1))
40		1re 11261	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ
41	40, 40	readdcli 11276	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
42	41	recni 11275	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) ∈ ℂ
43	4, 7	nn0addcld 12591	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℕ₀)
44	43	nn0red 12588	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℝ)
45	44	recnd 11289	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)) ∈ ℂ)
46		subeq0 11535	. . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → 1 ∈ ℂ)
50	49, 49, 30, 33	addsub4d 11667	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 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 11776	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
55	40, 5, 54	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐴)) ↔ (𝑝 pCnt 𝐴) ≤ 1))
56		subge0 11776	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
57	40, 8, 56	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (0 ≤ (1 − (𝑝 pCnt 𝐵)) ↔ (𝑝 pCnt 𝐵) ≤ 1))
58	55, 57	anbi12d 632	. . . . . . . . . . . . . 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 11573	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
60	40, 5, 59	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐴)) ∈ ℝ)
61		resubcl 11573	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐵) ∈ ℝ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
62	40, 8, 61	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (1 − (𝑝 pCnt 𝐵)) ∈ ℝ)
63		add20 11775	. . . . . . . . . . . . . . . . 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 660	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 2977	. . . . . . . . 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 12634	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
76		nnne0 12300	. . . . . . . . . . 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 12634	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
80		nnne0 12300	. . . . . . . . . . 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 16889	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt (𝐴 · 𝐵)) = ((𝑝 pCnt 𝐴) + (𝑝 pCnt 𝐵)))
85	84	breq1d 5153	. . . . . . 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 12542	. . . . . . . 8 ⊢ 1 ∈ ℕ₀
87		nn0leltp1 12677	. . . . . . . 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 586	. . . . . . 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 11362	. . . . . . . 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 586	. . . . . . 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 3167	. . . 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 27179	. . . . 5 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1))
96		issqf 27179	. . . . 5 ⊢ (𝐵 ∈ ℕ → ((μ‘𝐵) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐵) ≤ 1))
97	95, 96	bi2anan9 638	. . . 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 1133	. . 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 12290	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) ∈ ℕ)
100	99	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ (𝐴 gcd 𝐵) = 1) → (𝐴 · 𝐵) ∈ ℕ)
101		issqf 27179	. . . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ifcif 4525 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 gcd cgcd 16531 ℙcprime 16708 pCnt cpc 16874 μcmu 27138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-mu 27144

This theorem is referenced by: mumul 27224

W3C validator