ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pcqdiv Unicode version
Theorem pcqdiv 12845
Description: Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)
Assertion
Ref Expression
pcqdiv |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Proof of Theorem pcqdiv
StepHypRef Expression
1 simp2l 1047 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A e. QQ )
2 qcn 9841 . . . . . . 7 |- ( A e. QQ -> A e. CC )
31, 2syl 14 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A e. CC )
4 simp3l 1049 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B e. QQ )
5 qcn 9841 . . . . . . 7 |- ( B e. QQ -> B e. CC )
64, 5syl 14 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B e. CC )
7 simp3r 1050 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B =/= 0 )
8 0z 9468 . . . . . . . . 9 |- 0 e. ZZ
9 zq 9833 . . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. QQ )
108, 9ax-mp 5 . . . . . . . 8 |- 0 e. QQ
11 qapne 9846 . . . . . . . 8 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
124, 10, 11sylancl 413 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( B # 0 <-> B =/= 0 ) )
137, 12mpbird 167 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B # 0 )
143, 6, 13divcanap1d 8949 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) x. B ) = A )
1514oveq2d 6023 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( ( A / B ) x. B ) ) = ( P pCnt A ) )
16 simp1 1021 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> P e. Prime )
17 qdivcl 9850 . . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
181, 4, 7, 17syl3anc 1271 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) e. QQ )
19 simp2r 1048 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A =/= 0 )
20 qapne 9846 . . . . . . . . 9 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
211, 10, 20sylancl 413 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A # 0 <-> A =/= 0 ) )
2219, 21mpbird 167 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A # 0 )
233, 6, 22, 13divap0d 8964 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) # 0 )
24 qapne 9846 . . . . . . 7 |- ( ( ( A / B ) e. QQ /\ 0 e. QQ ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
2518, 10, 24sylancl 413 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
2623, 25mpbid 147 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) =/= 0 )
27 pcqmul 12841 . . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( ( A / B ) x. B ) ) = ( ( P pCnt ( A / B ) ) + ( P pCnt B ) ) )
2816, 18, 26, 4, 7, 27syl122anc 1280 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( ( A / B ) x. B ) ) = ( ( P pCnt ( A / B ) ) + ( P pCnt B ) ) )
2915, 28eqtr3d 2264 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt A ) = ( ( P pCnt ( A / B ) ) + ( P pCnt B ) ) )
3029oveq1d 6022 . 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( P pCnt A ) - ( P pCnt B ) ) = ( ( ( P pCnt ( A / B ) ) + ( P pCnt B ) ) - ( P pCnt B ) ) )
31 pcqcl 12844 . . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3216, 18, 26, 31syl12anc 1269 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3332zcnd 9581 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. CC )
34 pcqcl 12844 . . . . 5 |- ( ( P e. Prime /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
35343adant2 1040 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
3635zcnd 9581 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. CC )
3733, 36pncand 8469 . 2 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( ( P pCnt ( A / B ) ) + ( P pCnt B ) ) - ( P pCnt B ) ) = ( P pCnt ( A / B ) ) )
3830, 37eqtr2d 2263 1 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) = ( ( P pCnt A ) - ( P pCnt B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083  (class class class)co 6007   CCcc 8008   0cc0 8010   + caddc 8013   x. cmul 8015   - cmin 8328   # cap 8739   / cdiv 8830   ZZcz 9457   QQcq 9826   Primecprime 12644   pCnt cpc 12822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-dvds 12314  df-gcd 12490  df-prm 12645  df-pc 12823
This theorem is referenced by:  pcrec  12846
  Copyright terms: Public domain W3C validator