sqf11 - Metamath Proof Explorer

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Theorem sqf11 24665

Description: A squarefree number is completely determined by the set of its prime divisors. (Contributed by Mario Carneiro, 1-Jul-2015.)

**Assertion**
Ref	Expression
sqf11	⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))

Distinct variable groups: 𝐴,𝑝 𝐵,𝑝

**Proof of Theorem sqf11**
Step	Hyp	Ref	Expression
1		nnnn0 11176	. . . 4 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
2		nnnn0 11176	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℕ₀)
3		pc11 15422	. . . 4 ⊢ ((𝐴 ∈ ℕ₀ ∧ 𝐵 ∈ ℕ₀) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
4	1, 2, 3	syl2an 493	. . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
5	4	ad2ant2r 779	. 2 ⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
6		eleq1 2676	. . . . 5 ⊢ ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) → ((𝑝 pCnt 𝐴) ∈ ℕ ↔ (𝑝 pCnt 𝐵) ∈ ℕ))
7		dfbi3 933	. . . . . 6 ⊢ (((𝑝 pCnt 𝐴) ∈ ℕ ↔ (𝑝 pCnt 𝐵) ∈ ℕ) ↔ (((𝑝 pCnt 𝐴) ∈ ℕ ∧ (𝑝 pCnt 𝐵) ∈ ℕ) ∨ (¬ (𝑝 pCnt 𝐴) ∈ ℕ ∧ ¬ (𝑝 pCnt 𝐵) ∈ ℕ)))
8		simpll 786	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → 𝐴 ∈ ℕ)
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝐴 ∈ ℕ)
10		simpllr 795	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (μ‘𝐴) ≠ 0)
11		simpr 476	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝑝 ∈ ℙ)
12		sqfpc 24663	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0 ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ 1)
13	9, 10, 11, 12	syl3anc 1318	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ≤ 1)
14		nnle1eq1 10925	. . . . . . . . . 10 ⊢ ((𝑝 pCnt 𝐴) ∈ ℕ → ((𝑝 pCnt 𝐴) ≤ 1 ↔ (𝑝 pCnt 𝐴) = 1))
15	13, 14	syl5ibcom 234	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) ∈ ℕ → (𝑝 pCnt 𝐴) = 1))
16		simprl 790	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → 𝐵 ∈ ℕ)
17	16	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → 𝐵 ∈ ℕ)
18		simplrr 797	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (μ‘𝐵) ≠ 0)
19		sqfpc 24663	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0 ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ≤ 1)
20	17, 18, 11, 19	syl3anc 1318	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ≤ 1)
21		nnle1eq1 10925	. . . . . . . . . 10 ⊢ ((𝑝 pCnt 𝐵) ∈ ℕ → ((𝑝 pCnt 𝐵) ≤ 1 ↔ (𝑝 pCnt 𝐵) = 1))
22	20, 21	syl5ibcom 234	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐵) ∈ ℕ → (𝑝 pCnt 𝐵) = 1))
23	15, 22	anim12d 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ∈ ℕ ∧ (𝑝 pCnt 𝐵) ∈ ℕ) → ((𝑝 pCnt 𝐴) = 1 ∧ (𝑝 pCnt 𝐵) = 1)))
24		eqtr3 2631	. . . . . . . 8 ⊢ (((𝑝 pCnt 𝐴) = 1 ∧ (𝑝 pCnt 𝐵) = 1) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵))
25	23, 24	syl6 34	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ∈ ℕ ∧ (𝑝 pCnt 𝐵) ∈ ℕ) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
26		id 22	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℙ)
27		pccl 15392	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
28	26, 8, 27	syl2anr 494	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ₀)
29		elnn0 11171	. . . . . . . . . . 11 ⊢ ((𝑝 pCnt 𝐴) ∈ ℕ₀ ↔ ((𝑝 pCnt 𝐴) ∈ ℕ ∨ (𝑝 pCnt 𝐴) = 0))
30	28, 29	sylib 207	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) ∈ ℕ ∨ (𝑝 pCnt 𝐴) = 0))
31	30	ord 391	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (¬ (𝑝 pCnt 𝐴) ∈ ℕ → (𝑝 pCnt 𝐴) = 0))
32		pccl 15392	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑝 pCnt 𝐵) ∈ ℕ₀)
33	26, 16, 32	syl2anr 494	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐵) ∈ ℕ₀)
34		elnn0 11171	. . . . . . . . . . 11 ⊢ ((𝑝 pCnt 𝐵) ∈ ℕ₀ ↔ ((𝑝 pCnt 𝐵) ∈ ℕ ∨ (𝑝 pCnt 𝐵) = 0))
35	33, 34	sylib 207	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐵) ∈ ℕ ∨ (𝑝 pCnt 𝐵) = 0))
36	35	ord 391	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (¬ (𝑝 pCnt 𝐵) ∈ ℕ → (𝑝 pCnt 𝐵) = 0))
37	31, 36	anim12d 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((¬ (𝑝 pCnt 𝐴) ∈ ℕ ∧ ¬ (𝑝 pCnt 𝐵) ∈ ℕ) → ((𝑝 pCnt 𝐴) = 0 ∧ (𝑝 pCnt 𝐵) = 0)))
38		eqtr3 2631	. . . . . . . 8 ⊢ (((𝑝 pCnt 𝐴) = 0 ∧ (𝑝 pCnt 𝐵) = 0) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵))
39	37, 38	syl6 34	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((¬ (𝑝 pCnt 𝐴) ∈ ℕ ∧ ¬ (𝑝 pCnt 𝐵) ∈ ℕ) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
40	25, 39	jaod 394	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((((𝑝 pCnt 𝐴) ∈ ℕ ∧ (𝑝 pCnt 𝐵) ∈ ℕ) ∨ (¬ (𝑝 pCnt 𝐴) ∈ ℕ ∧ ¬ (𝑝 pCnt 𝐵) ∈ ℕ)) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
41	7, 40	syl5bi 231	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ∈ ℕ ↔ (𝑝 pCnt 𝐵) ∈ ℕ) → (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵)))
42	6, 41	impbid2 215	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ((𝑝 pCnt 𝐴) ∈ ℕ ↔ (𝑝 pCnt 𝐵) ∈ ℕ)))
43		pcelnn 15412	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝 pCnt 𝐴) ∈ ℕ ↔ 𝑝 ∥ 𝐴))
44	26, 8, 43	syl2anr 494	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) ∈ ℕ ↔ 𝑝 ∥ 𝐴))
45		pcelnn 15412	. . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → ((𝑝 pCnt 𝐵) ∈ ℕ ↔ 𝑝 ∥ 𝐵))
46	26, 16, 45	syl2anr 494	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐵) ∈ ℕ ↔ 𝑝 ∥ 𝐵))
47	44, 46	bibi12d 334	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → (((𝑝 pCnt 𝐴) ∈ ℕ ↔ (𝑝 pCnt 𝐵) ∈ ℕ) ↔ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))
48	42, 47	bitrd 267	. . 3 ⊢ ((((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) ∧ 𝑝 ∈ ℙ) → ((𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))
49	48	ralbidva 2968	. 2 ⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) = (𝑝 pCnt 𝐵) ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))
50	5, 49	bitrd 267	1 ⊢ (((𝐴 ∈ ℕ ∧ (μ‘𝐴) ≠ 0) ∧ (𝐵 ∈ ℕ ∧ (μ‘𝐵) ≠ 0)) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ↔ 𝑝 ∥ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 ≤ cle 9954 ℕcn 10897 ℕ₀cn0 11169 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 μcmu 24621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-mu 24627

This theorem is referenced by: (None)

W3C validator