Theorem pcdvdstr 12254

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 9198	. . . . . . 7 |- 0 e. ZZ
2		zq 9560	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
3	1, 2	ax-mp 5	. . . . . 6 |- 0 e. QQ
4		pcxcl 12239	. . . . . 6 |- ( ( P e. Prime /\ 0 e. QQ ) -> ( P pCnt 0 ) e. RR* )
5	3, 4	mpan2 422	. . . . 5 |- ( P e. Prime -> ( P pCnt 0 ) e. RR* )
6	5	xrleidd 9733	. . . 4 |- ( P e. Prime -> ( P pCnt 0 ) <_ ( P pCnt 0 ) )
7	6	ad2antrr 480	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt 0 ) <_ ( P pCnt 0 ) )
8		simpr 109	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A = 0 )
9	8	oveq2d 5857	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) = ( P pCnt 0 ) )
10		simplr3 1031	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A || B )
11	8, 10	eqbrtrrd 4005	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> 0 || B )
12		simplr2 1030	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B e. ZZ )
13		0dvds 11747	. . . . . 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 146	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B = 0 )
16	15	oveq2d 5857	. . 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 4013	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
18		prmnn 12038	. . . . . . 7 |- ( P e. Prime -> P e. NN )
19	18	ad2antrr 480	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. NN )
20		simpll 519	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. Prime )
21		simplr1 1029	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A e. ZZ )
22		simpr 109	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A =/= 0 )
23		pczcl 12226	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
24	20, 21, 22, 23	syl12anc 1226	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) e. NN0 )
25	19, 24	nnexpcld 10606	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. NN )
26	25	nnzd 9308	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
27		simplr2 1030	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> B e. ZZ )
28		pczdvds 12241	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
29	20, 21, 22, 28	syl12anc 1226	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || A )
30		simplr3 1031	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A || B )
31	26, 21, 27, 29, 30	dvdstrd 11766	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || B )
32		pcdvdsb 12247	. . . 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 1228	. . 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 166	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
35		simpr1 993	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> A e. ZZ )
36		zdceq 9262	. . . 4 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A = 0 )
37	35, 1, 36	sylancl 410	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> DECID A = 0 )
38		dcne 2346	. . 3 |- (DECID A = 0 <-> ( A = 0 \/ A =/= 0 ) )
39	37, 38	sylib 121	. 2 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( A = 0 \/ A =/= 0 ) )
40	17, 34, 39	mpjaodan 788	1 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> ( P pCnt A ) <_ ( P pCnt B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   class class class wbr 3981  (class class class)co 5841   0cc0 7749   RR*cxr 7928   <_ cle 7930   NNcn 8853   NN0cn0 9110   ZZcz 9187   QQcq 9553   ^cexp 10450   || cdvds 11723   Primecprime 12035   pCnt cpc 12212

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-1o 6380  df-2o 6381  df-er 6497  df-en 6703  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-fz 9941  df-fzo 10074  df-fl 10201  df-mod 10254  df-seqfrec 10377  df-exp 10451  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-dvds 11724  df-gcd 11872  df-prm 12036  df-pc 12213

This theorem is referenced by:  pcgcd1  12255  pc2dvds  12257