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Theorem pcqdiv 12235
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 1013 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A e. QQ )
2 qcn 9568 . . . . . . 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 1015 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B e. QQ )
5 qcn 9568 . . . . . . 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 1016 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B =/= 0 )
8 0z 9198 . . . . . . . . 9 |- 0 e. ZZ
9 zq 9560 . . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. QQ )
108, 9ax-mp 5 . . . . . . . 8 |- 0 e. QQ
11 qapne 9573 . . . . . . . 8 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
124, 10, 11sylancl 410 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( B # 0 <-> B =/= 0 ) )
137, 12mpbird 166 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B # 0 )
143, 6, 13divcanap1d 8683 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) x. B ) = A )
1514oveq2d 5857 . . . 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 987 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> P e. Prime )
17 qdivcl 9577 . . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
181, 4, 7, 17syl3anc 1228 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) e. QQ )
19 simp2r 1014 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A =/= 0 )
20 qapne 9573 . . . . . . . . 9 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
211, 10, 20sylancl 410 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A # 0 <-> A =/= 0 ) )
2219, 21mpbird 166 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A # 0 )
233, 6, 22, 13divap0d 8698 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) # 0 )
24 qapne 9573 . . . . . . 7 |- ( ( ( A / B ) e. QQ /\ 0 e. QQ ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
2518, 10, 24sylancl 410 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
2623, 25mpbid 146 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) =/= 0 )
27 pcqmul 12231 . . . . 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 1237 . . . 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 2200 . . 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 5856 . 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 12234 . . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3216, 18, 26, 31syl12anc 1226 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3332zcnd 9310 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. CC )
34 pcqcl 12234 . . . . 5 |- ( ( P e. Prime /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
35343adant2 1006 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
3635zcnd 9310 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. CC )
3733, 36pncand 8206 . 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 2199 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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   class class class wbr 3981  (class class class)co 5841   CCcc 7747   0cc0 7749   + caddc 7752   x. cmul 7754   - cmin 8065   # cap 8475   / cdiv 8564   ZZcz 9187   QQcq 9553   Primecprime 12035   pCnt cpc 12212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-en 6703  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872  df-prm 12036  df-pc 12213
This theorem is referenced by:  pcrec  12236
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