Theorem pcqcl 12878

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 9855	. . 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 9150	. . . . . . . . . . . 12 |- ( y e. NN -> y e. CC )
5		nnap0 9171	. . . . . . . . . . . 12 |- ( y e. NN -> y # 0 )
6	4, 5	div0apd 8966	. . . . . . . . . . 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 6024	. . . . . . . . . . 11 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
9	8	eqeq1d 2240	. . . . . . . . . 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 2446	. . . . . . . 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 12874	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) = ( ( P pCnt x ) - ( P pCnt y ) ) )
14		pczcl 12870	. . . . . . . . . . . . . . 15 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. NN0 )
15	14	nn0zd 9599	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( P pCnt x ) e. ZZ )
16	15	3adant3 1043	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt x ) e. ZZ )
17		nnz 9497	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y e. ZZ )
18		nnne0 9170	. . . . . . . . . . . . . . . 16 |- ( y e. NN -> y =/= 0 )
19	17, 18	jca 306	. . . . . . . . . . . . . . 15 |- ( y e. NN -> ( y e. ZZ /\ y =/= 0 ) )
20		pczcl 12870	. . . . . . . . . . . . . . . 16 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( P pCnt y ) e. NN0 )
21	20	nn0zd 9599	. . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt y ) e. ZZ )
24	16, 23	zsubcld 9606	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( ( P pCnt x ) - ( P pCnt y ) ) e. ZZ )
25	13, 24	eqeltrd 2308	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ y e. NN ) -> ( P pCnt ( x / y ) ) e. ZZ )
26	25	3expb 1230	. . . . . . . . . 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 2415	. . . . . . . 8 |- ( N = ( x / y ) -> ( N =/= 0 <-> ( x / y ) =/= 0 ) )
31		oveq2 6025	. . . . . . . . 9 |- ( N = ( x / y ) -> ( P pCnt N ) = ( P pCnt ( x / y ) ) )
32	31	eleq1d 2300	. . . . . . . 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 2657	. 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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511  (class class class)co 6017   0cc0 8031   - cmin 8349   / cdiv 8851   NNcn 9142   ZZcz 9478   QQcq 9852   Primecprime 12678   pCnt cpc 12856

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-sup 7182  df-inf 7183  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524  df-prm 12679  df-pc 12857

This theorem is referenced by:  pcqdiv  12879  pcexp  12881  pcxcl  12883  pcxqcl  12884  pcadd  12912  qexpz  12924  expnprm  12925