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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 27193 | . . 3 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) | |
2 | 1 | necon3abid 2975 | . 2 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
3 | ralnex 3070 | . . 3 ⊢ (∀𝑝 ∈ ℙ ¬ (𝑝↑2) ∥ 𝐴 ↔ ¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴) | |
4 | 1nn0 12540 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
5 | pccl 16883 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑝 pCnt 𝐴) ∈ ℕ0) | |
6 | 5 | ancoms 458 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℕ0) |
7 | nn0ltp1le 12674 | . . . . . . 7 ⊢ ((1 ∈ ℕ0 ∧ (𝑝 pCnt 𝐴) ∈ ℕ0) → (1 < (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴))) | |
8 | 4, 6, 7 | sylancr 587 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (1 < (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴))) |
9 | 1re 11259 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
10 | 6 | nn0red 12586 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝑝 pCnt 𝐴) ∈ ℝ) |
11 | ltnle 11338 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ (𝑝 pCnt 𝐴) ∈ ℝ) → (1 < (𝑝 pCnt 𝐴) ↔ ¬ (𝑝 pCnt 𝐴) ≤ 1)) | |
12 | 9, 10, 11 | sylancr 587 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (1 < (𝑝 pCnt 𝐴) ↔ ¬ (𝑝 pCnt 𝐴) ≤ 1)) |
13 | df-2 12327 | . . . . . . . 8 ⊢ 2 = (1 + 1) | |
14 | 13 | breq1i 5155 | . . . . . . 7 ⊢ (2 ≤ (𝑝 pCnt 𝐴) ↔ (1 + 1) ≤ (𝑝 pCnt 𝐴)) |
15 | id 22 | . . . . . . . 8 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℙ) | |
16 | nnz 12632 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
17 | 2nn0 12541 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
18 | pcdvdsb 16903 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 2 ∈ ℕ0) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) | |
19 | 17, 18 | mp3an3 1449 | . . . . . . . 8 ⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
20 | 15, 16, 19 | syl2anr 597 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (2 ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
21 | 14, 20 | bitr3id 285 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → ((1 + 1) ≤ (𝑝 pCnt 𝐴) ↔ (𝑝↑2) ∥ 𝐴)) |
22 | 8, 12, 21 | 3bitr3d 309 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (¬ (𝑝 pCnt 𝐴) ≤ 1 ↔ (𝑝↑2) ∥ 𝐴)) |
23 | 22 | con1bid 355 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (¬ (𝑝↑2) ∥ 𝐴 ↔ (𝑝 pCnt 𝐴) ≤ 1)) |
24 | 23 | ralbidva 3174 | . . 3 ⊢ (𝐴 ∈ ℕ → (∀𝑝 ∈ ℙ ¬ (𝑝↑2) ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
25 | 3, 24 | bitr3id 285 | . 2 ⊢ (𝐴 ∈ ℕ → (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝐴) ≤ 1)) |
26 | 2, 25 | bitrd 279 | 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 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 ↑cexp 14099 ∥ cdvds 16287 ℙcprime 16705 pCnt cpc 16870 μcmu 27153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fz 13545 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871 df-mu 27159 |
This theorem is referenced by: sqfpc 27195 mumullem2 27238 sqff1o 27240 |
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