Theorem pcqcl 12829

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 529	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2		elq 9817	. . 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 9118	. . . . . . . . . . . 12 |- ( y e. NN -> y e. CC )
5		nnap0 9139	. . . . . . . . . . . 12 |- ( y e. NN -> y # 0 )
6	4, 5	div0apd 8934	. . . . . . . . . . 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 6008	. . . . . . . . . . 11 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
9	8	eqeq1d 2238	. . . . . . . . . 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 2444	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( ( x / y ) =/= 0 -> x =/= 0 ) )
12		an32 562	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. NN ) /\ x =/= 0 ) <-> ( ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) )
13		pcdiv 12825	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
14		pczcl 12821	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. NN0 )
15	14	nn0zd 9567	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. ZZ )
16	15	3adant3 1041	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt x ) e. ZZ )
17		nnz 9465	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y e. ZZ )
18		nnne0 9138	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
19	17, 18	jca 306	. . . . . . . . . . . . . . 15 |- ( y e. NN -> ( y e. ZZ /\ y =/= 0 ) )
20		pczcl 12821	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. NN0 )
21	20	nn0zd 9567	. . . . . . . . . . . . . . 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 1040	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
24	16, 23	zsubcld 9574	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( ( P pCnt x ) - ( P pCnt y ) ) e. ZZ )
25	13, 24	eqeltrd 2306	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) e. ZZ )
26	25	3expb 1228	. . . . . . . . . 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 2413	. . . . . . . 8 |- ( N = ( x / y ) -> ( N =/= 0 <-> ( x / y ) =/= 0 ) )
31		oveq2 6009	. . . . . . . . 9 |- ( N = ( x / y ) -> ( P pCnt N ) = ( P pCnt ( x / y ) ) )
32	31	eleq1d 2298	. . . . . . . 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 2655	. 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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509  (class class class)co 6001   0cc0 7999   - cmin 8317   / cdiv 8819   NNcn 9110   ZZcz 9446   QQcq 9814   Primecprime 12629   pCnt cpc 12807

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-sup 7151  df-inf 7152  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-fl 10490  df-mod 10545  df-seqfrec 10670  df-exp 10761  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-dvds 12299  df-gcd 12475  df-prm 12630  df-pc 12808

This theorem is referenced by:  pcqdiv  12830  pcexp  12832  pcxcl  12834  pcxqcl  12835  pcadd  12863  qexpz  12875  expnprm  12876