Theorem pcdvdstr 12465

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 9328	. . . . . . 7 |- 0 e. ZZ
2		zq 9691	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
3	1, 2	ax-mp 5	. . . . . 6 |- 0 e. QQ
4		pcxcl 12449	. . . . . 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 9867	. . . 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 5934	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) = ( P pCnt 0 ) )
10		simplr3 1043	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A || B )
11	8, 10	eqbrtrrd 4053	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> 0 || B )
12		simplr2 1042	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B e. ZZ )
13		0dvds 11954	. . . . . 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 5934	. . 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 4061	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
18		prmnn 12248	. . . . . . 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 1041	. . . . . . 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 12436	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
24	20, 21, 22, 23	syl12anc 1247	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) e. NN0 )
25	19, 24	nnexpcld 10766	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. NN )
26	25	nnzd 9438	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
27		simplr2 1042	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> B e. ZZ )
28		pczdvds 12452	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
29	20, 21, 22, 28	syl12anc 1247	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || A )
30		simplr3 1043	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A || B )
31	26, 21, 27, 29, 30	dvdstrd 11973	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || B )
32		pcdvdsb 12458	. . . 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 1249	. . 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 1005	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> A e. ZZ )
36		zdceq 9392	. . . 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 2375	. . 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 799	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 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029  (class class class)co 5918   0cc0 7872   RR*cxr 8053   <_ cle 8055   NNcn 8982   NN0cn0 9240   ZZcz 9317   QQcq 9684   ^cexp 10609   || cdvds 11930   Primecprime 12245   pCnt cpc 12422

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246  df-pc 12423

This theorem is referenced by:  pcgcd1  12466  pc2dvds  12468