Theorem pcdvdstr 12280

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 9223	. . . . . . 7 |- 0 e. ZZ
2		zq 9585	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
3	1, 2	ax-mp 5	. . . . . 6 |- 0 e. QQ
4		pcxcl 12265	. . . . . 6 |- ( ( P e. Prime /\ 0 e. QQ ) -> ( P pCnt 0 ) e. RR* )
5	3, 4	mpan2 423	. . . . 5 |- ( P e. Prime -> ( P pCnt 0 ) e. RR* )
6	5	xrleidd 9758	. . . 4 |- ( P e. Prime -> ( P pCnt 0 ) <_ ( P pCnt 0 ) )
7	6	ad2antrr 485	. . 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 5869	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) = ( P pCnt 0 ) )
10		simplr3 1036	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A || B )
11	8, 10	eqbrtrrd 4013	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> 0 || B )
12		simplr2 1035	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B e. ZZ )
13		0dvds 11773	. . . . . 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 5869	. . 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 4021	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
18		prmnn 12064	. . . . . . 7 |- ( P e. Prime -> P e. NN )
19	18	ad2antrr 485	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. NN )
20		simpll 524	. . . . . . 7 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> P e. Prime )
21		simplr1 1034	. . . . . . 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 12252	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
24	20, 21, 22, 23	syl12anc 1231	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) e. NN0 )
25	19, 24	nnexpcld 10631	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. NN )
26	25	nnzd 9333	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
27		simplr2 1035	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> B e. ZZ )
28		pczdvds 12267	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
29	20, 21, 22, 28	syl12anc 1231	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || A )
30		simplr3 1036	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A || B )
31	26, 21, 27, 29, 30	dvdstrd 11792	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || B )
32		pcdvdsb 12273	. . . 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 1233	. . 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 998	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> A e. ZZ )
36		zdceq 9287	. . . 4 |- ( ( A e. ZZ /\ 0 e. ZZ ) -> DECID A = 0 )
37	35, 1, 36	sylancl 411	. . 3 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> DECID A = 0 )
38		dcne 2351	. . 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 793	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 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   class class class wbr 3989  (class class class)co 5853   0cc0 7774   RR*cxr 7953   <_ cle 7955   NNcn 8878   NN0cn0 9135   ZZcz 9212   QQcq 9578   ^cexp 10475   || cdvds 11749   Primecprime 12061   pCnt cpc 12238

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239

This theorem is referenced by:  pcgcd1  12281  pc2dvds  12283