Theorem pcmul 12819

Description: Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
pcmul	|- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )

Proof of Theorem pcmul

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2229	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
2		eqid 2229	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
3		eqid 2229	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < )
4	1, 2, 3	pcpremul 12811	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) + sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) = sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < ) )
5	1	pczpre 12815	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
6	5	3adant3 1041	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
7	2	pczpre 12815	. . . 4 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
8	7	3adant2 1040	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
9	6, 8	oveq12d 6018	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( P pCnt A ) + ( P pCnt B ) ) = ( sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) + sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) ) )
10		zmulcl 9496	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) e. ZZ )
11	10	ad2ant2r 509	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) e. ZZ )
12		zcn 9447	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
13	12	ad2antrr 488	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. CC )
14		zcn 9447	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
15	14	ad2antrl 490	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. CC )
16		simplr 528	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A =/= 0 )
17		simpll 527	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. ZZ )
18		0zd 9454	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> 0 e. ZZ )
19		zapne 9517	. . . . . . . . 9 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A # 0 <-> A =/= 0 ) )
20	17, 18, 19	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A # 0 <-> A =/= 0 ) )
21	16, 20	mpbird 167	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A # 0 )
22		simprr 531	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B =/= 0 )
23		simprl 529	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. ZZ )
24		zapne 9517	. . . . . . . . 9 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> ( B # 0 <-> B =/= 0 ) )
25	23, 18, 24	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( B # 0 <-> B =/= 0 ) )
26	22, 25	mpbird 167	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B # 0 )
27	13, 15, 21, 26	mulap0d 8801	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) # 0 )
28		zapne 9517	. . . . . . 7 |- ( ( ( A x. B ) e. ZZ /\ 0 e. ZZ ) -> ( ( A x. B ) # 0 <-> ( A x. B ) =/= 0 ) )
29	11, 18, 28	syl2anc 411	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( A x. B ) # 0 <-> ( A x. B ) =/= 0 ) )
30	27, 29	mpbid 147	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) =/= 0 )
31	11, 30	jca 306	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( A x. B ) e. ZZ /\ ( A x. B ) =/= 0 ) )
32	3	pczpre 12815	. . . 4 |- ( ( P e. Prime /\ ( ( A x. B ) e. ZZ /\ ( A x. B ) =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < ) )
33	31, 32	sylan2 286	. . 3 |- ( ( P e. Prime /\ ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) ) -> ( P pCnt ( A x. B ) ) = sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < ) )
34	33	3impb 1223	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = sup ( { n e. NN0 | ( P ^ n ) || ( A x. B ) } , RR , < ) )
35	4, 9, 34	3eqtr4rd 2273	1 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt ( A x. B ) ) = ( ( P pCnt A ) + ( P pCnt B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   class class class wbr 4082  (class class class)co 6000   supcsup 7145   CCcc 7993   RRcr 7994   0cc0 7995   + caddc 7998   x. cmul 8000   < clt 8177   # cap 8724   NN0cn0 9365   ZZcz 9442   ^cexp 10755   || cdvds 12293   Primecprime 12624   pCnt cpc 12802

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-fl 10485  df-mod 10540  df-seqfrec 10665  df-exp 10756  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-dvds 12294  df-gcd 12470  df-prm 12625  df-pc 12803

This theorem is referenced by:  pcqmul  12821  pcaddlem  12857  pcmpt  12861  pcfac  12868  pcbc  12869  lgsdi  15710