Theorem pczpre 12297

Description: Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
pczpre.1	|- S = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )

Assertion

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

Distinct variable groups:   n, N   P, n

Allowed substitution hint:   S( n)

Proof of Theorem pczpre

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

Step	Hyp	Ref	Expression
1		zq 9626	. . 3 |- ( N e. ZZ -> N e. QQ )
2		eqid 2177	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3		eqid 2177	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4	2, 3	pcval 12296	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) = ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
5	1, 4	sylanr1 404	. 2 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
6		simprl 529	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
7	6	zcnd 9376	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
8	7	div1d 8737	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / 1 ) = N )
9	8	eqcomd 2183	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N = ( N / 1 ) )
10		prmuz2 12131	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
11		eqid 2177	. . . . . . . 8 |- 1 = 1
12		eqid 2177	. . . . . . . . 9 |- { n e. NN0 | ( P ^ n ) || 1 } = { n e. NN0 | ( P ^ n ) || 1 }
13		eqid 2177	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < )
14	12, 13	pcpre1 12292	. . . . . . . 8 |- ( ( P e. ( ZZ>= ` 2 ) /\ 1 = 1 ) -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
15	10, 11, 14	sylancl 413	. . . . . . 7 |- ( P e. Prime -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
16	15	adantr 276	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
17	16	oveq2d 5891	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) = ( S - 0 ) )
18		eqid 2177	. . . . . . . . . 10 |- { n e. NN0 | ( P ^ n ) || N } = { n e. NN0 | ( P ^ n ) || N }
19		pczpre.1	. . . . . . . . . 10 |- S = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )
20	18, 19	pcprecl 12289	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
21	10, 20	sylan 283	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
22	21	simpld 112	. . . . . . 7 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
23	22	nn0cnd 9231	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. CC )
24	23	subid1d 8257	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - 0 ) = S )
25	17, 24	eqtr2d 2211	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
26		1nn 8930	. . . . 5 |- 1 e. NN
27		oveq1 5882	. . . . . . . 8 |- ( x = N -> ( x / y ) = ( N / y ) )
28	27	eqeq2d 2189	. . . . . . 7 |- ( x = N -> ( N = ( x / y ) <-> N = ( N / y ) ) )
29		breq2 4008	. . . . . . . . . . . 12 |- ( x = N -> ( ( P ^ n ) || x <-> ( P ^ n ) || N ) )
30	29	rabbidv 2727	. . . . . . . . . . 11 |- ( x = N -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || N } )
31	30	supeq1d 6986	. . . . . . . . . 10 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) )
32	31, 19	eqtr4di 2228	. . . . . . . . 9 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = S )
33	32	oveq1d 5890	. . . . . . . 8 |- ( x = N -> ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) )
34	33	eqeq2d 2189	. . . . . . 7 |- ( x = N -> ( S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
35	28, 34	anbi12d 473	. . . . . 6 |- ( x = N -> ( ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( N = ( N / y ) /\ S = ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
36		oveq2 5883	. . . . . . . 8 |- ( y = 1 -> ( N / y ) = ( N / 1 ) )
37	36	eqeq2d 2189	. . . . . . 7 |- ( y = 1 -> ( N = ( N / y ) <-> N = ( N / 1 ) ) )
38		breq2 4008	. . . . . . . . . . 11 |- ( y = 1 -> ( ( P ^ n ) || y <-> ( P ^ n ) || 1 ) )
39	38	rabbidv 2727	. . . . . . . . . 10 |- ( y = 1 -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || 1 } )
40	39	supeq1d 6986	. . . . . . . . 9 |- ( y = 1 -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) )
41	40	oveq2d 5891	. . . . . . . 8 |- ( y = 1 -> ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
42	41	eqeq2d 2189	. . . . . . 7 |- ( y = 1 -> ( S = ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) ) )
43	37, 42	anbi12d 473	. . . . . 6 |- ( y = 1 -> ( ( N = ( N / y ) /\ S = ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( N = ( N / 1 ) /\ S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) ) ) )
44	35, 43	rspc2ev 2857	. . . . 5 |- ( ( N e. ZZ /\ 1 e. NN /\ ( N = ( N / 1 ) /\ S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) ) ) -> E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
45	26, 44	mp3an2 1325	. . . 4 |- ( ( N e. ZZ /\ ( N = ( N / 1 ) /\ S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) ) ) -> E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
46	6, 9, 25, 45	syl12anc 1236	. . 3 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
47		reex 7945	. . . . . 6 |- RR e. _V
48		supex2g 7032	. . . . . 6 |- ( RR e. _V -> sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) e. _V )
49	47, 48	ax-mp 5	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) e. _V
50	19, 49	eqeltri 2250	. . . 4 |- S e. _V
51	2, 3	pceu 12295	. . . . 5 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E! z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
52	1, 51	sylanr1 404	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> E! z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
53		eqeq1 2184	. . . . . . 7 |- ( z = S -> ( z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) <-> S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) )
54	53	anbi2d 464	. . . . . 6 |- ( z = S -> ( ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
55	54	2rexbidv 2502	. . . . 5 |- ( z = S -> ( E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) )
56	55	iota2 5207	. . . 4 |- ( ( S e. _V /\ E! z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) -> ( E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = S ) )
57	50, 52, 56	sylancr 414	. . 3 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ S = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) <-> ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = S ) )
58	46, 57	mpbid 147	. 2 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) ) ) = S )
59	5, 58	eqtrd 2210	1 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E!weu 2026   e. wcel 2148   =/= wne 2347   E.wrex 2456   {crab 2459   _Vcvv 2738   class class class wbr 4004   iotacio 5177   ` cfv 5217  (class class class)co 5875   supcsup 6981   RRcr 7810   0cc0 7811   1c1 7812   < clt 7992   - cmin 8128   / cdiv 8629   NNcn 8919   2c2 8970   NN0cn0 9176   ZZcz 9253   ZZ>=cuz 9528   QQcq 9619   ^cexp 10519   || cdvds 11794   Primecprime 12107   pCnt cpc 12284

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-1o 6417  df-2o 6418  df-er 6535  df-en 6741  df-sup 6983  df-inf 6984  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944  df-prm 12108  df-pc 12285

This theorem is referenced by:  pczcl  12298  pcmul  12301  pcdiv  12302  pc1  12305  pczdvds  12313  pczndvds  12315  pczndvds2  12317  pcneg  12324