Theorem pczpre 12820

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 9821	. . 3 |- ( N e. ZZ -> N e. QQ )
2		eqid 2229	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3		eqid 2229	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4	2, 3	pcval 12819	. . 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 9570	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
8	7	div1d 8927	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / 1 ) = N )
9	8	eqcomd 2235	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N = ( N / 1 ) )
10		prmuz2 12653	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
11		eqid 2229	. . . . . . . 8 |- 1 = 1
12		eqid 2229	. . . . . . . . 9 |- { n e. NN0 | ( P ^ n ) || 1 } = { n e. NN0 | ( P ^ n ) || 1 }
13		eqid 2229	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < )
14	12, 13	pcpre1 12815	. . . . . . . 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 6017	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) = ( S - 0 ) )
18		eqid 2229	. . . . . . . . . 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 12812	. . . . . . . . 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 9424	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. CC )
24	23	subid1d 8446	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - 0 ) = S )
25	17, 24	eqtr2d 2263	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
26		1nn 9121	. . . . 5 |- 1 e. NN
27		oveq1 6008	. . . . . . . 8 |- ( x = N -> ( x / y ) = ( N / y ) )
28	27	eqeq2d 2241	. . . . . . 7 |- ( x = N -> ( N = ( x / y ) <-> N = ( N / y ) ) )
29		breq2 4087	. . . . . . . . . . . 12 |- ( x = N -> ( ( P ^ n ) || x <-> ( P ^ n ) || N ) )
30	29	rabbidv 2788	. . . . . . . . . . 11 |- ( x = N -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || N } )
31	30	supeq1d 7154	. . . . . . . . . 10 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) )
32	31, 19	eqtr4di 2280	. . . . . . . . 9 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = S )
33	32	oveq1d 6016	. . . . . . . 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 2241	. . . . . . 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 6009	. . . . . . . 8 |- ( y = 1 -> ( N / y ) = ( N / 1 ) )
37	36	eqeq2d 2241	. . . . . . 7 |- ( y = 1 -> ( N = ( N / y ) <-> N = ( N / 1 ) ) )
38		breq2 4087	. . . . . . . . . . 11 |- ( y = 1 -> ( ( P ^ n ) || y <-> ( P ^ n ) || 1 ) )
39	38	rabbidv 2788	. . . . . . . . . 10 |- ( y = 1 -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || 1 } )
40	39	supeq1d 7154	. . . . . . . . 9 |- ( y = 1 -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) )
41	40	oveq2d 6017	. . . . . . . 8 |- ( y = 1 -> ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
42	41	eqeq2d 2241	. . . . . . 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 2922	. . . . 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 1359	. . . 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 1269	. . 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 8133	. . . . . 6 |- RR e. _V
48		supex2g 7200	. . . . . 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 2302	. . . 4 |- S e. _V
51	2, 3	pceu 12818	. . . . 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 2236	. . . . . . 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 2555	. . . . 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 5308	. . . 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 2262	1 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E!weu 2077   e. wcel 2200   =/= wne 2400   E.wrex 2509   {crab 2512   _Vcvv 2799   class class class wbr 4083   iotacio 5276   ` cfv 5318  (class class class)co 6001   supcsup 7149   RRcr 7998   0cc0 7999   1c1 8000   < clt 8181   - cmin 8317   / cdiv 8819   NNcn 9110   2c2 9161   NN0cn0 9369   ZZcz 9446   ZZ>=cuz 9722   QQcq 9814   ^cexp 10760   || cdvds 12298   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:  pczcl  12821  pcmul  12824  pcdiv  12825  pc1  12828  pczdvds  12837  pczndvds  12839  pczndvds2  12841  pcneg  12848