Theorem pcdvdstr 12871

Description: The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)

Assertion

Ref	Expression
pcdvdstr	|- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )

Proof of Theorem pcdvdstr

Step	Hyp	Ref	Expression
1		0z 9473	. . . . . . 7 |- 0 e. ZZ
2		zq 9838	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
3	1, 2	ax-mp 5	. . . . . 6 |- 0 e. QQ
4		pcxcl 12855	. . . . . 6 |- ( ( P e. Prime /\ 0 e. QQ ) -> ( P pCnt 0 ) e. RR* )
5	3, 4	mpan2 425	. . . . 5 |- ( P e. Prime -> ( P pCnt 0 ) e. RR* )
6	5	xrleidd 10014	. . . 4 |- ( P e. Prime -> ( P pCnt 0 ) <_ ( P pCnt 0 ) )
7	6	ad2antrr 488	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt 0 ) <_ ( P pCnt 0 ) )
8		simpr 110	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A = 0 )
9	8	oveq2d 6026	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) = ( P pCnt 0 ) )
10		simplr3 1065	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A || B )
11	8, 10	eqbrtrrd 4107	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> 0 || B )
12		simplr2 1064	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B e. ZZ )
13		0dvds 12343	. . . . . 6 |- ( B e. ZZ -> ( 0 || B <-> B = 0 ) )
14	12, 13	syl 14	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( 0 || B <-> B = 0 ) )
15	11, 14	mpbid 147	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B = 0 )
16	15	oveq2d 6026	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt B ) = ( P pCnt 0 ) )
17	7, 9, 16	3brtr4d 4115	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
18		prmnn 12653	. . . . . . 7 |- ( P e. Prime -> P e. NN )
19	18	ad2antrr 488	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. NN )
20		simpll 527	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. Prime )
21		simplr1 1063	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A e. ZZ )
22		simpr 110	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A =/= 0 )
23		pczcl 12842	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
24	20, 21, 22, 23	syl12anc 1269	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) e. NN0 )
25	19, 24	nnexpcld 10934	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. NN )
26	25	nnzd 9584	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
27		simplr2 1064	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> B e. ZZ )
28		pczdvds 12858	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
29	20, 21, 22, 28	syl12anc 1269	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || A )
30		simplr3 1065	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A || B )
31	26, 21, 27, 29, 30	dvdstrd 12362	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || B )
32		pcdvdsb 12864	. . . 4 |- ( ( P e. Prime /\ B e. ZZ /\ ( P pCnt A ) e. NN0 ) -> ( ( P pCnt A ) <_ ( P pCnt B ) <-> ( P ^ ( P pCnt A ) ) || B ) )
33	20, 27, 24, 32	syl3anc 1271	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( ( P pCnt A ) <_ ( P pCnt B ) <-> ( P ^ ( P pCnt A ) ) || B ) )
34	31, 33	mpbird 167	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
35		simpr1 1027	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> A e. ZZ )
36		zdceq 9538	. . . 4 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A = 0 )
37	35, 1, 36	sylancl 413	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> DECID A = 0 )
38		dcne 2411	. . 3 |- (DECID A = 0 <-> ( A = 0 \/ A =/= 0 ) )
39	37, 38	sylib 122	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( A = 0 \/ A =/= 0 ) )
40	17, 34, 39	mpjaodan 803	1 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083  (class class class)co 6010   0cc0 8015   RR*cxr 8196   <_ cle 8198   NNcn 9126   NN0cn0 9385   ZZcz 9462   QQcq 9831   ^cexp 10777   || cdvds 12319   Primecprime 12650   pCnt cpc 12828

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-stab 836  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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-2o 6574  df-er 6693  df-en 6901  df-sup 7167  df-inf 7168  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-fl 10507  df-mod 10562  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-dvds 12320  df-gcd 12496  df-prm 12651  df-pc 12829

This theorem is referenced by:  pcgcd1  12872  pc2dvds  12874  dvdsppwf1o  15684