Theorem pczpre 12653

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 9749	. . 3 |- ( N e. ZZ -> N e. QQ )
2		eqid 2205	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3		eqid 2205	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4	2, 3	pcval 12652	. . 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 9498	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
8	7	div1d 8855	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / 1 ) = N )
9	8	eqcomd 2211	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N = ( N / 1 ) )
10		prmuz2 12486	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
11		eqid 2205	. . . . . . . 8 |- 1 = 1
12		eqid 2205	. . . . . . . . 9 |- { n e. NN0 | ( P ^ n ) || 1 } = { n e. NN0 | ( P ^ n ) || 1 }
13		eqid 2205	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < )
14	12, 13	pcpre1 12648	. . . . . . . 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 5962	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) = ( S - 0 ) )
18		eqid 2205	. . . . . . . . . 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 12645	. . . . . . . . 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 9352	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. CC )
24	23	subid1d 8374	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - 0 ) = S )
25	17, 24	eqtr2d 2239	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
26		1nn 9049	. . . . 5 |- 1 e. NN
27		oveq1 5953	. . . . . . . 8 |- ( x = N -> ( x / y ) = ( N / y ) )
28	27	eqeq2d 2217	. . . . . . 7 |- ( x = N -> ( N = ( x / y ) <-> N = ( N / y ) ) )
29		breq2 4049	. . . . . . . . . . . 12 |- ( x = N -> ( ( P ^ n ) || x <-> ( P ^ n ) || N ) )
30	29	rabbidv 2761	. . . . . . . . . . 11 |- ( x = N -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || N } )
31	30	supeq1d 7091	. . . . . . . . . 10 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) )
32	31, 19	eqtr4di 2256	. . . . . . . . 9 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = S )
33	32	oveq1d 5961	. . . . . . . 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 2217	. . . . . . 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 5954	. . . . . . . 8 |- ( y = 1 -> ( N / y ) = ( N / 1 ) )
37	36	eqeq2d 2217	. . . . . . 7 |- ( y = 1 -> ( N = ( N / y ) <-> N = ( N / 1 ) ) )
38		breq2 4049	. . . . . . . . . . 11 |- ( y = 1 -> ( ( P ^ n ) || y <-> ( P ^ n ) || 1 ) )
39	38	rabbidv 2761	. . . . . . . . . 10 |- ( y = 1 -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || 1 } )
40	39	supeq1d 7091	. . . . . . . . 9 |- ( y = 1 -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) )
41	40	oveq2d 5962	. . . . . . . 8 |- ( y = 1 -> ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
42	41	eqeq2d 2217	. . . . . . 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 2892	. . . . 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 1338	. . . 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 1248	. . 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 8061	. . . . . 6 |- RR e. _V
48		supex2g 7137	. . . . . 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 2278	. . . 4 |- S e. _V
51	2, 3	pceu 12651	. . . . 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 2212	. . . . . . 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 2531	. . . . 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 5262	. . . 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 2238	1 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E!weu 2054   e. wcel 2176   =/= wne 2376   E.wrex 2485   {crab 2488   _Vcvv 2772   class class class wbr 4045   iotacio 5231   ` cfv 5272  (class class class)co 5946   supcsup 7086   RRcr 7926   0cc0 7927   1c1 7928   < clt 8109   - cmin 8245   / cdiv 8747   NNcn 9038   2c2 9089   NN0cn0 9297   ZZcz 9374   ZZ>=cuz 9650   QQcq 9742   ^cexp 10685   || cdvds 12131   Primecprime 12462   pCnt cpc 12640

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-fz 10133  df-fzo 10267  df-fl 10415  df-mod 10470  df-seqfrec 10595  df-exp 10686  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-dvds 12132  df-gcd 12308  df-prm 12463  df-pc 12641

This theorem is referenced by:  pczcl  12654  pcmul  12657  pcdiv  12658  pc1  12661  pczdvds  12670  pczndvds  12672  pczndvds2  12674  pcneg  12681