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Theorem pcqdiv 13064
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 1054 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A e. QQ )
2 qcn 10013 . . . . . . 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 1056 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B e. QQ )
5 qcn 10013 . . . . . . 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 1057 . . . . . . 7 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> B =/= 0 )
8 0z 9634 . . . . . . . . 9 |- 0 e. ZZ
9 zq 10005 . . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. QQ )
108, 9ax-mp 5 . . . . . . . 8 |- 0 e. QQ
11 qapne 10018 . . . . . . . 8 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
124, 10, 11sylancl 417 . . . . . . 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 9111 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A / B ) x. B ) = A )
1514oveq2d 6091 . . . 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 1028 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> P e. Prime )
17 qdivcl 10022 . . . . . 6 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
181, 4, 7, 17syl3anc 1278 . . . . 5 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) e. QQ )
19 simp2r 1055 . . . . . . . 8 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> A =/= 0 )
20 qapne 10018 . . . . . . . . 9 |- ( ( A e. QQ /\ 0 e. QQ ) -> ( A # 0 <-> A =/= 0 ) )
211, 10, 20sylancl 417 . . . . . . . 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 9126 . . . . . 6 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A / B ) # 0 )
24 qapne 10018 . . . . . . 7 |- ( ( ( A / B ) e. QQ /\ 0 e. QQ ) -> ( ( A / B ) # 0 <-> ( A / B ) =/= 0 ) )
2518, 10, 24sylancl 417 . . . . . 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 13060 . . . . 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 1287 . . . 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 2273 . . 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 6090 . 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 13063 . . . . 5 |- ( ( P e. Prime /\ ( ( A / B ) e. QQ /\ ( A / B ) =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3216, 18, 26, 31syl12anc 1276 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. ZZ )
3332zcnd 9748 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt ( A / B ) ) e. CC )
34 pcqcl 13063 . . . . 5 |- ( ( P e. Prime /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
35343adant2 1047 . . . 4 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. ZZ )
3635zcnd 9748 . . 3 |- ( ( P e. Prime /\ ( A e. QQ /\ A =/= 0 ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( P pCnt B ) e. CC )
3733, 36pncand 8628 . 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 2272 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   class class class wbr 4125  (class class class)co 6075   CCcc 8167   0cc0 8169   + caddc 8172   x. cmul 8174   - cmin 8487   # cap 8899   / cdiv 8992   ZZcz 9623   QQcq 9998   Primecprime 12863   pCnt cpc 13041
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-dvds 12533  df-gcd 12709  df-prm 12864  df-pc 13042
This theorem is referenced by:  pcrec  13065
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