Theorem pcmul 12318

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 2188	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
2		eqid 2188	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
3		eqid 2188	. . 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 12310	. 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 12314	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
6	5	3adant3 1018	. . 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 12314	. . . 4 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
8	7	3adant2 1017	. . 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 5908	. 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 9323	. . . . . 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 9275	. . . . . . . 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 9275	. . . . . . . 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 9282	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> 0 e. ZZ )
19		zapne 9344	. . . . . . . . 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 9344	. . . . . . . . 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 8632	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) # 0 )
28		zapne 9344	. . . . . . 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 12314	. . . 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 1200	. 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 2232	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 979   = wceq 1363   e. wcel 2159   =/= wne 2359   {crab 2471   class class class wbr 4017  (class class class)co 5890   supcsup 6998   CCcc 7826   RRcr 7827   0cc0 7828   + caddc 7831   x. cmul 7833   < clt 8009   # cap 8555   NN0cn0 9193   ZZcz 9270   ^cexp 10536   || cdvds 11811   Primecprime 12124   pCnt cpc 12301

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947  ax-caucvg 7948

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-if 3549  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-ilim 4383  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-isom 5239  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-frec 6409  df-1o 6434  df-2o 6435  df-er 6552  df-en 6758  df-sup 7000  df-inf 7001  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-2 8995  df-3 8996  df-4 8997  df-n0 9194  df-z 9271  df-uz 9546  df-q 9637  df-rp 9671  df-fz 10026  df-fzo 10160  df-fl 10287  df-mod 10340  df-seqfrec 10463  df-exp 10537  df-cj 10868  df-re 10869  df-im 10870  df-rsqrt 11024  df-abs 11025  df-dvds 11812  df-gcd 11961  df-prm 12125  df-pc 12302

This theorem is referenced by:  pcqmul  12320  pcaddlem  12355  pcmpt  12358  pcfac  12365  pcbc  12366  lgsdi  14821