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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 | β’ (π΄ β β β ((ΞΌβπ΄) = 0 β βπ β β (πβ2) β₯ π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12327 | . . . . . 6 β’ -1 β β | |
2 | neg1ne0 12329 | . . . . . 6 β’ -1 β 0 | |
3 | prmdvdsfi 26990 | . . . . . . . 8 β’ (π΄ β β β {π β β β£ π β₯ π΄} β Fin) | |
4 | hashcl 14319 | . . . . . . . 8 β’ ({π β β β£ π β₯ π΄} β Fin β (β―β{π β β β£ π β₯ π΄}) β β0) | |
5 | 3, 4 | syl 17 | . . . . . . 7 β’ (π΄ β β β (β―β{π β β β£ π β₯ π΄}) β β0) |
6 | 5 | nn0zd 12585 | . . . . . 6 β’ (π΄ β β β (β―β{π β β β£ π β₯ π΄}) β β€) |
7 | expne0i 14063 | . . . . . 6 β’ ((-1 β β β§ -1 β 0 β§ (β―β{π β β β£ π β₯ π΄}) β β€) β (-1β(β―β{π β β β£ π β₯ π΄})) β 0) | |
8 | 1, 2, 6, 7 | mp3an12i 1461 | . . . . 5 β’ (π΄ β β β (-1β(β―β{π β β β£ π β₯ π΄})) β 0) |
9 | iffalse 4532 | . . . . . 6 β’ (Β¬ βπ β β (πβ2) β₯ π΄ β if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) = (-1β(β―β{π β β β£ π β₯ π΄}))) | |
10 | 9 | neeq1d 2994 | . . . . 5 β’ (Β¬ βπ β β (πβ2) β₯ π΄ β (if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) β 0 β (-1β(β―β{π β β β£ π β₯ π΄})) β 0)) |
11 | 8, 10 | syl5ibrcom 246 | . . . 4 β’ (π΄ β β β (Β¬ βπ β β (πβ2) β₯ π΄ β if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) β 0)) |
12 | muval 27015 | . . . . 5 β’ (π΄ β β β (ΞΌβπ΄) = if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄})))) | |
13 | 12 | neeq1d 2994 | . . . 4 β’ (π΄ β β β ((ΞΌβπ΄) β 0 β if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) β 0)) |
14 | 11, 13 | sylibrd 259 | . . 3 β’ (π΄ β β β (Β¬ βπ β β (πβ2) β₯ π΄ β (ΞΌβπ΄) β 0)) |
15 | 14 | necon4bd 2954 | . 2 β’ (π΄ β β β ((ΞΌβπ΄) = 0 β βπ β β (πβ2) β₯ π΄)) |
16 | iftrue 4529 | . . 3 β’ (βπ β β (πβ2) β₯ π΄ β if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) = 0) | |
17 | 12 | eqeq1d 2728 | . . 3 β’ (π΄ β β β ((ΞΌβπ΄) = 0 β if(βπ β β (πβ2) β₯ π΄, 0, (-1β(β―β{π β β β£ π β₯ π΄}))) = 0)) |
18 | 16, 17 | imbitrrid 245 | . 2 β’ (π΄ β β β (βπ β β (πβ2) β₯ π΄ β (ΞΌβπ΄) = 0)) |
19 | 15, 18 | impbid 211 | 1 β’ (π΄ β β β ((ΞΌβπ΄) = 0 β βπ β β (πβ2) β₯ π΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1533 β wcel 2098 β wne 2934 βwrex 3064 {crab 3426 ifcif 4523 class class class wbr 5141 βcfv 6536 (class class class)co 7404 Fincfn 8938 βcc 11107 0cc0 11109 1c1 11110 -cneg 11446 βcn 12213 2c2 12268 β0cn0 12473 β€cz 12559 βcexp 14030 β―chash 14293 β₯ cdvds 16202 βcprime 16613 ΞΌcmu 26978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-exp 14031 df-hash 14294 df-dvds 16203 df-prm 16614 df-mu 26984 |
This theorem is referenced by: issqf 27019 dvdssqf 27021 mumullem1 27062 |
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