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Mirrors > Home > MPE Home > Th. List > issqf | Structured version Visualization version GIF version |
Description: Two ways to say that a number is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
issqf | ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnsqf 25213 | . . 3 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) | |
2 | 1 | necon3abid 3007 | . 2 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
3 | ralnex 3173 | . . 3 ⊢ (∀𝑝 ∈ ℙ ¬ (𝑝↑2) ∥ 𝐴 ↔ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) | |
4 | 1nn0 11598 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | pccl 15887 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑝 pCnt 𝐴) ∈ ℕ0) | |
6 | 5 | ancoms 451 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ0) |
7 | nn0ltp1le 11725 | . . . . . . 7 ⊢ ((1 ∈ ℕ0 ∧ (𝑝 pCnt 𝐴) ∈ ℕ0) → (1 < (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴))) | |
8 | 4, 6, 7 | sylancr 582 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (1 < (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴))) |
9 | 1re 10328 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | 6 | nn0red 11641 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ) |
11 | ltnle 10407 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (1 < (𝑝 pCnt 𝐴) ↔ ¬ (𝑝 pCnt 𝐴) ≤ 1)) | |
12 | 9, 10, 11 | sylancr 582 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (1 < (𝑝 pCnt 𝐴) ↔ ¬ (𝑝 pCnt 𝐴) ≤ 1)) |
13 | df-2 11376 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
14 | 13 | breq1i 4850 | . . . . . . 7 ⊢ (2 ≤ (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴)) |
15 | id 22 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℙ) | |
16 | nnz 11689 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
17 | 2nn0 11599 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
18 | pcdvdsb 15906 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ0) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) | |
19 | 17, 18 | mp3an3 1575 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
20 | 15, 16, 19 | syl2anr 591 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
21 | 14, 20 | syl5bbr 277 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → ((1 + 1) ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
22 | 8, 12, 21 | 3bitr3d 301 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (¬ (𝑝 pCnt 𝐴) ≤ 1 ↔ (𝑝↑2) ∥ 𝐴)) |
23 | 22 | con1bid 347 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (¬ (𝑝↑2) ∥ 𝐴 ↔ (𝑝 pCnt 𝐴) ≤ 1)) |
24 | 23 | ralbidva 3166 | . . 3 ⊢ (𝐴 ∈ ℕ → (∀𝑝 ∈ ℙ ¬ (𝑝↑2) ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
25 | 3, 24 | syl5bbr 277 | . 2 ⊢ (𝐴 ∈ ℕ → (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
26 | 2, 25 | bitrd 271 | 1 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∃wrex 3090 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 ℝcr 10223 0cc0 10224 1c1 10225 + caddc 10227 < clt 10363 ≤ cle 10364 ℕcn 11312 2c2 11368 ℕ0cn0 11580 ℤcz 11666 ↑cexp 13114 ∥ cdvds 15319 ℙcprime 15719 pCnt cpc 15874 μcmu 25173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-fz 12581 df-fl 12848 df-mod 12924 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-dvds 15320 df-gcd 15552 df-prm 15720 df-pc 15875 df-mu 25179 |
This theorem is referenced by: sqfpc 25215 mumullem2 25258 sqff1o 25260 |
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