Theorem pcmul 12497

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 2196	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < )
2		eqid 2196	. . 3 |- sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < )
3		eqid 2196	. . 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 12489	. 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 12493	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) = sup ( { n e. NN0 | ( P ^ n ) || A } , RR , < ) )
6	5	3adant3 1019	. . 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 12493	. . . 4 |- ( ( P e. Prime /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( P pCnt B ) = sup ( { n e. NN0 | ( P ^ n ) || B } , RR , < ) )
8	7	3adant2 1018	. . 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 5943	. 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 9398	. . . . . 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 9350	. . . . . . . 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 9350	. . . . . . . 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 9357	. . . . . . . . 9 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> 0 e. ZZ )
19		zapne 9419	. . . . . . . . 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 9419	. . . . . . . . 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 8704	. . . . . 6 |- ( ( ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( A x. B ) # 0 )
28		zapne 9419	. . . . . . 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 12493	. . . 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 1201	. 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 2240	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 980   = wceq 1364   e. wcel 2167   =/= wne 2367   {crab 2479   class class class wbr 4034  (class class class)co 5925   supcsup 7057   CCcc 7896   RRcr 7897   0cc0 7898   + caddc 7901   x. cmul 7903   < clt 8080   # cap 8627   NN0cn0 9268   ZZcz 9345   ^cexp 10649   || cdvds 11971   Primecprime 12302   pCnt cpc 12480

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-fz 10103  df-fzo 10237  df-fl 10379  df-mod 10434  df-seqfrec 10559  df-exp 10650  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-dvds 11972  df-gcd 12148  df-prm 12303  df-pc 12481

This theorem is referenced by:  pcqmul  12499  pcaddlem  12535  pcmpt  12539  pcfac  12546  pcbc  12547  lgsdi  15386