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Theorem pcqdiv 12339
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 1025 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A e. QQ )
2 qcn 9664 . . . . . . 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 1027 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B e. QQ )
5 qcn 9664 . . . . . . 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 1028 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B =/= 0 )
8 0z 9294 . . . . . . . . 9 |- 0 e. ZZ
9 zq 9656 . . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. QQ )
108, 9ax-mp 5 . . . . . . . 8 |- 0 e. QQ
11 qapne 9669 . . . . . . . 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 8778 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) x. B ) = A )
1514oveq2d 5912 . . . 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 999 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> P e. Prime )
17 qdivcl 9673 . . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
181, 4, 7, 17syl3anc 1249 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) e. QQ )
19 simp2r 1026 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A =/= 0 )
20 qapne 9669 . . . . . . . . 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 8793 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) # 0 )
24 qapne 9669 . . . . . . 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 12335 . . . . 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 1258 . . . 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 2224 . . 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 5911 . 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 12338 . . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3216, 18, 26, 31syl12anc 1247 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3332zcnd 9406 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. CC )
34 pcqcl 12338 . . . . 5 |- ( ( P e. Prime /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
35343adant2 1018 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
3635zcnd 9406 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. CC )
3733, 36pncand 8299 . 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 2223 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 980   = wceq 1364   e. wcel 2160   =/= wne 2360   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841   + caddc 7844   x. cmul 7846   - cmin 8158   # cap 8568   / cdiv 8659   ZZcz 9283   QQcq 9649   Primecprime 12139   pCnt cpc 12316
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-1o 6441  df-2o 6442  df-er 6559  df-en 6767  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-fz 10039  df-fzo 10173  df-fl 10301  df-mod 10354  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-dvds 11827  df-gcd 11976  df-prm 12140  df-pc 12317
This theorem is referenced by:  pcrec  12340
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