Theorem pcmul 13080

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 2238	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
2		eqid 2238	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
3		eqid 2238	. . 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 13072	. 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 13076	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
6	5	3adant3 1048	. . 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 13076	. . . 4 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
8	7	3adant2 1047	. . 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 6103	. 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 9698	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) e. ZZ )
11	10	ad2ant2r 513	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) e. ZZ )
12		zcn 9649	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
13	12	ad2antrr 492	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. CC )
14		zcn 9649	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
15	14	ad2antrl 494	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. CC )
16		simplr 533	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A =/= 0 )
17		simpll 531	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. ZZ )
18		0zd 9656	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> 0 e. ZZ )
19		zapne 9719	. . . . . . . . 9 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A # 0 <-> A =/= 0 ) )
20	17, 18, 19	syl2anc 415	. . . . . . . 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 537	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B =/= 0 )
23		simprl 535	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. ZZ )
24		zapne 9719	. . . . . . . . 9 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> ( B # 0 <-> B =/= 0 ) )
25	23, 18, 24	syl2anc 415	. . . . . . . 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 8986	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) # 0 )
28		zapne 9719	. . . . . . 7 |- ( ( ( A x. B ) e. ZZ /\ 0 e. ZZ ) -> ( ( A x. B ) # 0 <-> ( A x. B ) =/= 0 ) )
29	11, 18, 28	syl2anc 415	. . . . . 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 13076	. . . 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 1230	. 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 2282	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   class class class wbr 4130  (class class class)co 6085   supcsup 7322   CCcc 8177   RRcr 8178   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   # cap 8909   NN0cn0 9563   ZZcz 9644   ^cexp 10975   || cdvds 12554   Primecprime 12885   pCnt cpc 13063

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-pc 13064

This theorem is used by:  pcqmul  13082  pcaddlem  13118  pcmpt  13122  pcfac  13129  pcbc  13130  lgsdi  16156