Theorem pczpre 12491

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 9717	. . 3 |- ( N e. ZZ -> N e. QQ )
2		eqid 2196	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3		eqid 2196	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4	2, 3	pcval 12490	. . 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 9466	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
8	7	div1d 8824	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / 1 ) = N )
9	8	eqcomd 2202	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N = ( N / 1 ) )
10		prmuz2 12324	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
11		eqid 2196	. . . . . . . 8 |- 1 = 1
12		eqid 2196	. . . . . . . . 9 |- { n e. NN0 | ( P ^ n ) || 1 } = { n e. NN0 | ( P ^ n ) || 1 }
13		eqid 2196	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < )
14	12, 13	pcpre1 12486	. . . . . . . 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 5941	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) = ( S - 0 ) )
18		eqid 2196	. . . . . . . . . 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 12483	. . . . . . . . 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 9321	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. CC )
24	23	subid1d 8343	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - 0 ) = S )
25	17, 24	eqtr2d 2230	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
26		1nn 9018	. . . . 5 |- 1 e. NN
27		oveq1 5932	. . . . . . . 8 |- ( x = N -> ( x / y ) = ( N / y ) )
28	27	eqeq2d 2208	. . . . . . 7 |- ( x = N -> ( N = ( x / y ) <-> N = ( N / y ) ) )
29		breq2 4038	. . . . . . . . . . . 12 |- ( x = N -> ( ( P ^ n ) || x <-> ( P ^ n ) || N ) )
30	29	rabbidv 2752	. . . . . . . . . . 11 |- ( x = N -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || N } )
31	30	supeq1d 7062	. . . . . . . . . 10 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) )
32	31, 19	eqtr4di 2247	. . . . . . . . 9 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = S )
33	32	oveq1d 5940	. . . . . . . 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 2208	. . . . . . 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 5933	. . . . . . . 8 |- ( y = 1 -> ( N / y ) = ( N / 1 ) )
37	36	eqeq2d 2208	. . . . . . 7 |- ( y = 1 -> ( N = ( N / y ) <-> N = ( N / 1 ) ) )
38		breq2 4038	. . . . . . . . . . 11 |- ( y = 1 -> ( ( P ^ n ) || y <-> ( P ^ n ) || 1 ) )
39	38	rabbidv 2752	. . . . . . . . . 10 |- ( y = 1 -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || 1 } )
40	39	supeq1d 7062	. . . . . . . . 9 |- ( y = 1 -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) )
41	40	oveq2d 5941	. . . . . . . 8 |- ( y = 1 -> ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
42	41	eqeq2d 2208	. . . . . . 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 2883	. . . . 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 1336	. . . 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 1247	. . 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 8030	. . . . . 6 |- RR e. _V
48		supex2g 7108	. . . . . 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 2269	. . . 4 |- S e. _V
51	2, 3	pceu 12489	. . . . 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 2203	. . . . . . 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 2522	. . . . 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 5249	. . . 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 2229	1 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E!weu 2045   e. wcel 2167   =/= wne 2367   E.wrex 2476   {crab 2479   _Vcvv 2763   class class class wbr 4034   iotacio 5218   ` cfv 5259  (class class class)co 5925   supcsup 7057   RRcr 7895   0cc0 7896   1c1 7897   < clt 8078   - cmin 8214   / cdiv 8716   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   ZZ>=cuz 9618   QQcq 9710   ^cexp 10647   || cdvds 11969   Primecprime 12300   pCnt cpc 12478

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301  df-pc 12479

This theorem is referenced by:  pczcl  12492  pcmul  12495  pcdiv  12496  pc1  12499  pczdvds  12508  pczndvds  12510  pczndvds2  12512  pcneg  12519