Theorem pcdvdstr 12720

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 9398	. . . . . . 7 |- 0 e. ZZ
2		zq 9762	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
3	1, 2	ax-mp 5	. . . . . 6 |- 0 e. QQ
4		pcxcl 12704	. . . . . 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 9938	. . . 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 5972	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) = ( P pCnt 0 ) )
10		simplr3 1044	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> A || B )
11	8, 10	eqbrtrrd 4074	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> 0 || B )
12		simplr2 1043	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> B e. ZZ )
13		0dvds 12192	. . . . . 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 5972	. . 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 4082	. 2 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A = 0 ) -> ( P pCnt A ) <_ ( P pCnt B ) )
18		prmnn 12502	. . . . . . 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 1042	. . . . . . 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 12691	. . . . . . 7 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P pCnt A ) e. NN0 )
24	20, 21, 22, 23	syl12anc 1248	. . . . . 6 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P pCnt A ) e. NN0 )
25	19, 24	nnexpcld 10857	. . . . 5 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. NN )
26	25	nnzd 9509	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) e. ZZ )
27		simplr2 1043	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> B e. ZZ )
28		pczdvds 12707	. . . . 5 |- ( ( P e. Prime /\ ( A e. ZZ /\ A =/= 0 ) ) -> ( P ^ ( P pCnt A ) ) || A )
29	20, 21, 22, 28	syl12anc 1248	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || A )
30		simplr3 1044	. . . 4 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> A || B )
31	26, 21, 27, 29, 30	dvdstrd 12211	. . 3 |- ( ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) /\ A =/= 0 ) -> ( P ^ ( P pCnt A ) ) || B )
32		pcdvdsb 12713	. . . 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 1250	. . 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 1006	. . . 4 |- ( ( P e. Prime /\ ( A e. ZZ /\ B e. ZZ /\ A || B ) ) -> A e. ZZ )
36		zdceq 9463	. . . 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 2388	. . 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 800	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   class class class wbr 4050  (class class class)co 5956   0cc0 7940   RR*cxr 8121   <_ cle 8123   NNcn 9051   NN0cn0 9310   ZZcz 9387   QQcq 9755   ^cexp 10700   || cdvds 12168   Primecprime 12499   pCnt cpc 12677

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057  ax-pre-mulext 8058  ax-arch 8059  ax-caucvg 8060

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-isom 5288  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-1o 6514  df-2o 6515  df-er 6632  df-en 6840  df-sup 7100  df-inf 7101  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-div 8761  df-inn 9052  df-2 9110  df-3 9111  df-4 9112  df-n0 9311  df-z 9388  df-uz 9664  df-q 9756  df-rp 9791  df-fz 10146  df-fzo 10280  df-fl 10430  df-mod 10485  df-seqfrec 10610  df-exp 10701  df-cj 11223  df-re 11224  df-im 11225  df-rsqrt 11379  df-abs 11380  df-dvds 12169  df-gcd 12345  df-prm 12500  df-pc 12678

This theorem is referenced by:  pcgcd1  12721  pc2dvds  12723  dvdsppwf1o  15531