Theorem pcmul 12259

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 2171	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
2		eqid 2171	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
3		eqid 2171	. . 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 12251	. 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 12255	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
6	5	3adant3 1013	. . 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 12255	. . . 4 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
8	7	3adant2 1012	. . 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 5875	. 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 9269	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A x. B ) e. ZZ )
11	10	ad2ant2r 507	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) e. ZZ )
12		zcn 9221	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
13	12	ad2antrr 486	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. CC )
14		zcn 9221	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
15	14	ad2antrl 488	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. CC )
16		simplr 526	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A =/= 0 )
17		simpll 525	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A e. ZZ )
18		0zd 9228	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> 0 e. ZZ )
19		zapne 9290	. . . . . . . . 9 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> ( A # 0 <-> A =/= 0 ) )
20	17, 18, 19	syl2anc 409	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A # 0 <-> A =/= 0 ) )
21	16, 20	mpbird 166	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> A # 0 )
22		simprr 528	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B =/= 0 )
23		simprl 527	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B e. ZZ )
24		zapne 9290	. . . . . . . . 9 |- ( ( B e. ZZ /\ 0 e. ZZ ) -> ( B # 0 <-> B =/= 0 ) )
25	23, 18, 24	syl2anc 409	. . . . . . . 8 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( B # 0 <-> B =/= 0 ) )
26	22, 25	mpbird 166	. . . . . . 7 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> B # 0 )
27	13, 15, 21, 26	mulap0d 8580	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) # 0 )
28		zapne 9290	. . . . . . 7 |- ( ( ( A x. B ) e. ZZ /\ 0 e. ZZ ) -> ( ( A x. B ) # 0 <-> ( A x. B ) =/= 0 ) )
29	11, 18, 28	syl2anc 409	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( A x. B ) # 0 <-> ( A x. B ) =/= 0 ) )
30	27, 29	mpbid 146	. . . . 5 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) =/= 0 )
31	11, 30	jca 304	. . . 4 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( ( A x. B ) e. ZZ /\ ( A x. B ) =/= 0 ) )
32	3	pczpre 12255	. . . 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 284	. . 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 1195	. 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 2215	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 103   <-> wb 104   /\ w3a 974   = wceq 1349   e. wcel 2142   =/= wne 2341   {crab 2453   class class class wbr 3990  (class class class)co 5857   supcsup 6963   CCcc 7776   RRcr 7777   0cc0 7778   + caddc 7781   x. cmul 7783   < clt 7958   # cap 8504   NN0cn0 9139   ZZcz 9216   ^cexp 10479   || cdvds 11753   Primecprime 12065   pCnt cpc 12242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895  ax-pre-mulext 7896  ax-arch 7897  ax-caucvg 7898

This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-isom 5209  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-frec 6374  df-1o 6399  df-2o 6400  df-er 6517  df-en 6723  df-sup 6965  df-inf 6966  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-reap 8498  df-ap 8505  df-div 8594  df-inn 8883  df-2 8941  df-3 8942  df-4 8943  df-n0 9140  df-z 9217  df-uz 9492  df-q 9583  df-rp 9615  df-fz 9970  df-fzo 10103  df-fl 10230  df-mod 10283  df-seqfrec 10406  df-exp 10480  df-cj 10810  df-re 10811  df-im 10812  df-rsqrt 10966  df-abs 10967  df-dvds 11754  df-gcd 11902  df-prm 12066  df-pc 12243

This theorem is referenced by:  pcqmul  12261  pcaddlem  12296  pcmpt  12299  pcfac  12306  pcbc  12307  lgsdi  13817