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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 12357 | . . . . . 6 β’ -1 β β | |
2 | neg1ne0 12359 | . . . . . 6 β’ -1 β 0 | |
3 | prmdvdsfi 27052 | . . . . . . . 8 β’ (π΄ β β β {π β β β£ π β₯ π΄} β Fin) | |
4 | hashcl 14348 | . . . . . . . 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 12615 | . . . . . 6 β’ (π΄ β β β (β―β{π β β β£ π β₯ π΄}) β β€) |
7 | expne0i 14092 | . . . . . 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 1462 | . . . . 5 β’ (π΄ β β β (-1β(β―β{π β β β£ π β₯ π΄})) β 0) |
9 | iffalse 4538 | . . . . . 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 2997 | . . . . 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 27077 | . . . . 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 2997 | . . . 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 2957 | . 2 β’ (π΄ β β β ((ΞΌβπ΄) = 0 β βπ β β (πβ2) β₯ π΄)) |
16 | iftrue 4535 | . . 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 2730 | . . 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 1534 β wcel 2099 β wne 2937 βwrex 3067 {crab 3429 ifcif 4529 class class class wbr 5148 βcfv 6548 (class class class)co 7420 Fincfn 8964 βcc 11137 0cc0 11139 1c1 11140 -cneg 11476 βcn 12243 2c2 12298 β0cn0 12503 β€cz 12589 βcexp 14059 β―chash 14322 β₯ cdvds 16231 βcprime 16642 ΞΌcmu 27040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-seq 14000 df-exp 14060 df-hash 14323 df-dvds 16232 df-prm 16643 df-mu 27046 |
This theorem is referenced by: issqf 27081 dvdssqf 27083 mumullem1 27124 |
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