Theorem pcqcl 13004

Description: Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
pcqcl	|- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) e. ZZ )

Proof of Theorem pcqcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 531	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2		elq 9954	. . 3 |- ( N e. QQ <-> E. x e. ZZ E. y e. NN N = ( x / y ) )
3	1, 2	sylib 122	. 2 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E. x e. ZZ E. y e. NN N = ( x / y ) )
4		nncn 9245	. . . . . . . . . . . 12 |- ( y e. NN -> y e. CC )
5		nnap0 9266	. . . . . . . . . . . 12 |- ( y e. NN -> y # 0 )
6	4, 5	div0apd 9061	. . . . . . . . . . 11 |- ( y e. NN -> ( 0 / y ) = 0 )
7	6	ad2antll 491	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( 0 / y ) = 0 )
8		oveq1 6057	. . . . . . . . . . 11 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
9	8	eqeq1d 2241	. . . . . . . . . 10 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
10	7, 9	syl5ibrcom 157	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( x = 0 -> ( x / y ) = 0 ) )
11	10	necon3d 2456	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( ( x / y ) =/= 0 -> x =/= 0 ) )
12		an32 564	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. NN ) /\ x =/= 0 ) <-> ( ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) )
13		pcdiv 13000	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
14		pczcl 12996	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. NN0 )
15	14	nn0zd 9698	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. ZZ )
16	15	3adant3 1044	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt x ) e. ZZ )
17		nnz 9596	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y e. ZZ )
18		nnne0 9265	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
19	17, 18	jca 306	. . . . . . . . . . . . . . 15 |- ( y e. NN -> ( y e. ZZ /\ y =/= 0 ) )
20		pczcl 12996	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. NN0 )
21	20	nn0zd 9698	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. ZZ )
22	19, 21	sylan2 286	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
23	22	3adant2 1043	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
24	16, 23	zsubcld 9705	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( ( P pCnt x ) - ( P pCnt y ) ) e. ZZ )
25	13, 24	eqeltrd 2309	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) e. ZZ )
26	25	3expb 1231	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) ) -> ( P pCnt ( x / y ) ) e. ZZ )
27	12, 26	sylan2b 287	. . . . . . . . 9 |- ( ( P e. Prime /\ ( ( x e. ZZ /\ y e. NN ) /\ x =/= 0 ) ) -> ( P pCnt ( x / y ) ) e. ZZ )
28	27	expr 375	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( x =/= 0 -> ( P pCnt ( x / y ) ) e. ZZ ) )
29	11, 28	syld 45	. . . . . . 7 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( ( x / y ) =/= 0 -> ( P pCnt ( x / y ) ) e. ZZ ) )
30		neeq1 2425	. . . . . . . 8 |- ( N = ( x / y ) -> ( N =/= 0 <-> ( x / y ) =/= 0 ) )
31		oveq2 6058	. . . . . . . . 9 |- ( N = ( x / y ) -> ( P pCnt N ) = ( P pCnt ( x / y ) ) )
32	31	eleq1d 2301	. . . . . . . 8 |- ( N = ( x / y ) -> ( ( P pCnt N ) e. ZZ <-> ( P pCnt ( x / y ) ) e. ZZ ) )
33	30, 32	imbi12d 234	. . . . . . 7 |- ( N = ( x / y ) -> ( ( N =/= 0 -> ( P pCnt N ) e. ZZ ) <-> ( ( x / y ) =/= 0 -> ( P pCnt ( x / y ) ) e. ZZ ) ) )
34	29, 33	syl5ibrcom 157	. . . . . 6 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( N = ( x / y ) -> ( N =/= 0 -> ( P pCnt N ) e. ZZ ) ) )
35	34	com23 78	. . . . 5 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( N =/= 0 -> ( N = ( x / y ) -> ( P pCnt N ) e. ZZ ) ) )
36	35	impancom 260	. . . 4 |- ( ( P e. Prime /\ N =/= 0 ) -> ( ( x e. ZZ /\ y e. NN ) -> ( N = ( x / y ) -> ( P pCnt N ) e. ZZ ) ) )
37	36	adantrl 478	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( ( x e. ZZ /\ y e. NN ) -> ( N = ( x / y ) -> ( P pCnt N ) e. ZZ ) ) )
38	37	rexlimdvv 2667	. 2 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( E. x e. ZZ E. y e. NN N = ( x / y ) -> ( P pCnt N ) e. ZZ ) )
39	3, 38	mpd 13	1 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) e. ZZ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   E.wrex 2521  (class class class)co 6050   0cc0 8127   - cmin 8444   / cdiv 8946   NNcn 9237   ZZcz 9577   QQcq 9951   Primecprime 12804   pCnt cpc 12982

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805  df-pc 12983

This theorem is referenced by:  pcqdiv  13005  pcexp  13007  pcxcl  13009  pcxqcl  13010  pcadd  13038  qexpz  13050  expnprm  13051