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Mirrors > Home > MPE Home > Th. List > isnsqf | Structured version Visualization version GIF version |
Description: Two ways to say that a number is not squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.) |
Ref | Expression |
---|---|
isnsqf | ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmdvdsfi 25024 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → {𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin) | |
2 | hashcl 13331 | . . . . . . . 8 ⊢ ({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴} ∈ Fin → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ0) | |
3 | 1, 2 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℕ0) |
4 | 3 | nn0zd 11664 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ) |
5 | neg1cn 11308 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
6 | neg1ne0 11310 | . . . . . . 7 ⊢ -1 ≠ 0 | |
7 | expne0i 13078 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ -1 ≠ 0 ∧ (♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ) → (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ≠ 0) | |
8 | 5, 6, 7 | mp3an12 1555 | . . . . . 6 ⊢ ((♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}) ∈ ℤ → (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ≠ 0) |
9 | 4, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ≠ 0) |
10 | iffalse 4231 | . . . . . 6 ⊢ (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) | |
11 | 10 | neeq1d 2983 | . . . . 5 ⊢ (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → (if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) ≠ 0 ↔ (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})) ≠ 0)) |
12 | 9, 11 | syl5ibrcom 237 | . . . 4 ⊢ (𝐴 ∈ ℕ → (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) ≠ 0)) |
13 | muval 25049 | . . . . 5 ⊢ (𝐴 ∈ ℕ → (μ‘𝐴) = if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴})))) | |
14 | 13 | neeq1d 2983 | . . . 4 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) ≠ 0 ↔ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) ≠ 0)) |
15 | 12, 14 | sylibrd 249 | . . 3 ⊢ (𝐴 ∈ ℕ → (¬ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → (μ‘𝐴) ≠ 0)) |
16 | 15 | necon4bd 2944 | . 2 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 → ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
17 | iftrue 4228 | . . 3 ⊢ (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0) | |
18 | 13 | eqeq1d 2754 | . . 3 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ if(∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴, 0, (-1↑(♯‘{𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐴}))) = 0)) |
19 | 17, 18 | syl5ibr 236 | . 2 ⊢ (𝐴 ∈ ℕ → (∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴 → (μ‘𝐴) = 0)) |
20 | 16, 19 | impbid 202 | 1 ⊢ (𝐴 ∈ ℕ → ((μ‘𝐴) = 0 ↔ ∃𝑝 ∈ ℙ (𝑝↑2) ∥ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ∃wrex 3043 {crab 3046 ifcif 4222 class class class wbr 4796 ‘cfv 6041 (class class class)co 6805 Fincfn 8113 ℂcc 10118 0cc0 10120 1c1 10121 -cneg 10451 ℕcn 11204 2c2 11254 ℕ0cn0 11476 ℤcz 11561 ↑cexp 13046 ♯chash 13303 ∥ cdvds 15174 ℙcprime 15579 μcmu 25012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-card 8947 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-n0 11477 df-z 11562 df-uz 11872 df-fz 12512 df-seq 12988 df-exp 13047 df-hash 13304 df-dvds 15175 df-prm 15580 df-mu 25018 |
This theorem is referenced by: issqf 25053 dvdssqf 25055 mumullem1 25096 |
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