Theorem pcqcl 12234

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 521	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
2		elq 9556	. . 3 |- ( N e. QQ <-> E. x e. ZZ E. y e. NN N = ( x / y ) )
3	1, 2	sylib 121	. 2 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E. x e. ZZ E. y e. NN N = ( x / y ) )
4		nncn 8861	. . . . . . . . . . . 12 |- ( y e. NN -> y e. CC )
5		nnap0 8882	. . . . . . . . . . . 12 |- ( y e. NN -> y # 0 )
6	4, 5	div0apd 8679	. . . . . . . . . . 11 |- ( y e. NN -> ( 0 / y ) = 0 )
7	6	ad2antll 483	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( 0 / y ) = 0 )
8		oveq1 5848	. . . . . . . . . . 11 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
9	8	eqeq1d 2174	. . . . . . . . . 10 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
10	7, 9	syl5ibrcom 156	. . . . . . . . 9 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( x = 0 -> ( x / y ) = 0 ) )
11	10	necon3d 2379	. . . . . . . 8 |- ( ( P e. Prime /\ ( x e. ZZ /\ y e. NN ) ) -> ( ( x / y ) =/= 0 -> x =/= 0 ) )
12		an32 552	. . . . . . . . . 10 |- ( ( ( x e. ZZ /\ y e. NN ) /\ x =/= 0 ) <-> ( ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) )
13		pcdiv 12230	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
14		pczcl 12226	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. NN0 )
15	14	nn0zd 9307	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. ZZ )
16	15	3adant3 1007	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt x ) e. ZZ )
17		nnz 9206	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y e. ZZ )
18		nnne0 8881	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
19	17, 18	jca 304	. . . . . . . . . . . . . . 15 |- ( y e. NN -> ( y e. ZZ /\ y =/= 0 ) )
20		pczcl 12226	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. NN0 )
21	20	nn0zd 9307	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. ZZ )
22	19, 21	sylan2 284	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
23	22	3adant2 1006	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
24	16, 23	zsubcld 9314	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( ( P pCnt x ) - ( P pCnt y ) ) e. ZZ )
25	13, 24	eqeltrd 2242	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) e. ZZ )
26	25	3expb 1194	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) ) -> ( P pCnt ( x / y ) ) e. ZZ )
27	12, 26	sylan2b 285	. . . . . . . . 9 |- ( ( P e. Prime /\ ( ( x e. ZZ /\ y e. NN ) /\ x =/= 0 ) ) -> ( P pCnt ( x / y ) ) e. ZZ )
28	27	expr 373	. . . . . . . 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 2348	. . . . . . . 8 |- ( N = ( x / y ) -> ( N =/= 0 <-> ( x / y ) =/= 0 ) )
31		oveq2 5849	. . . . . . . . 9 |- ( N = ( x / y ) -> ( P pCnt N ) = ( P pCnt ( x / y ) ) )
32	31	eleq1d 2234	. . . . . . . 8 |- ( N = ( x / y ) -> ( ( P pCnt N ) e. ZZ <-> ( P pCnt ( x / y ) ) e. ZZ ) )
33	30, 32	imbi12d 233	. . . . . . 7 |- ( N = ( x / y ) -> ( ( N =/= 0 -> ( P pCnt N ) e. ZZ ) <-> ( ( x / y ) =/= 0 -> ( P pCnt ( x / y ) ) e. ZZ ) ) )
34	29, 33	syl5ibrcom 156	. . . . . 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 258	. . . 4 |- ( ( P e. Prime /\ N =/= 0 ) -> ( ( x e. ZZ /\ y e. NN ) -> ( N = ( x / y ) -> ( P pCnt N ) e. ZZ ) ) )
37	36	adantrl 470	. . 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 2589	. 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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2335   E.wrex 2444  (class class class)co 5841   0cc0 7749   - cmin 8065   / cdiv 8564   NNcn 8853   ZZcz 9187   QQcq 9553   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-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:  pcqdiv  12235  pcexp  12237  pcxcl  12239  pcadd  12267  qexpz  12278  expnprm  12279