Theorem pczpre 15552

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 11794	. . 3 |- ( N e. ZZ -> N e. QQ )
2		eqid 2622	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
3		eqid 2622	. . . 4 |- sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
4	2, 3	pcval 15549	. . 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 684	. 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 794	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
7	6	zcnd 11483	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
8	7	div1d 10793	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / 1 ) = N )
9	8	eqcomd 2628	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> N = ( N / 1 ) )
10		prmuz2 15408	. . . . . . . 8 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
11		eqid 2622	. . . . . . . 8 |- 1 = 1
12		eqid 2622	. . . . . . . . 9 |- { n e. NN0 | ( P ^ n ) || 1 } = { n e. NN0 | ( P ^ n ) || 1 }
13		eqid 2622	. . . . . . . . 9 |- sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < )
14	12, 13	pcpre1 15547	. . . . . . . 8 |- ( ( P e. ( ZZ>= ` 2 ) /\ 1 = 1 ) -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
15	10, 11, 14	sylancl 694	. . . . . . 7 |- ( P e. Prime -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
16	15	adantr 481	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) = 0 )
17	16	oveq2d 6666	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) = ( S - 0 ) )
18		eqid 2622	. . . . . . . . . 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 15544	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
21	10, 20	sylan 488	. . . . . . . 8 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || N ) )
22	21	simpld 475	. . . . . . 7 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
23	22	nn0cnd 11353	. . . . . 6 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. CC )
24	23	subid1d 10381	. . . . 5 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S - 0 ) = S )
25	17, 24	eqtr2d 2657	. . . 4 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> S = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
26		1nn 11031	. . . . 5 |- 1 e. NN
27		oveq1 6657	. . . . . . . 8 |- ( x = N -> ( x / y ) = ( N / y ) )
28	27	eqeq2d 2632	. . . . . . 7 |- ( x = N -> ( N = ( x / y ) <-> N = ( N / y ) ) )
29		breq2 4657	. . . . . . . . . . . 12 |- ( x = N -> ( ( P ^ n ) || x <-> ( P ^ n ) || N ) )
30	29	rabbidv 3189	. . . . . . . . . . 11 |- ( x = N -> { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || N } )
31	30	supeq1d 8352	. . . . . . . . . 10 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) )
32	31, 19	syl6eqr 2674	. . . . . . . . 9 |- ( x = N -> sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) = S )
33	32	oveq1d 6665	. . . . . . . 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 2632	. . . . . . 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 747	. . . . . 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 6658	. . . . . . . 8 |- ( y = 1 -> ( N / y ) = ( N / 1 ) )
37	36	eqeq2d 2632	. . . . . . 7 |- ( y = 1 -> ( N = ( N / y ) <-> N = ( N / 1 ) ) )
38		breq2 4657	. . . . . . . . . . 11 |- ( y = 1 -> ( ( P ^ n ) || y <-> ( P ^ n ) || 1 ) )
39	38	rabbidv 3189	. . . . . . . . . 10 |- ( y = 1 -> { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || 1 } )
40	39	supeq1d 8352	. . . . . . . . 9 |- ( y = 1 -> sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) )
41	40	oveq2d 6666	. . . . . . . 8 |- ( y = 1 -> ( S - sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) ) = ( S - sup ( { n e. NN0 | ( P ^ n ) || 1 } , RR , < ) ) )
42	41	eqeq2d 2632	. . . . . . 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 747	. . . . . 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 3324	. . . . 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 1412	. . . 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 1324	. . 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		ltso 10118	. . . . . 6 |- < Or RR
48	47	supex 8369	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < ) e. _V
49	19, 48	eqeltri 2697	. . . 4 |- S e. _V
50	2, 3	pceu 15551	. . . . 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 , < ) ) ) )
51	1, 50	sylanr1 684	. . . 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 , < ) ) ) )
52		eqeq1 2626	. . . . . . 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 , < ) ) ) )
53	52	anbi2d 740	. . . . . 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 , < ) ) ) ) )
54	53	2rexbidv 3057	. . . . 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 , < ) ) ) ) )
55	54	iota2 5877	. . . 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 ) )
56	49, 51, 55	sylancr 695	. . 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 ) )
57	46, 56	mpbid 222	. 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 )
58	5, 57	eqtrd 2656	1 |- ( ( P e. Prime /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P pCnt N ) = S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   iotacio 5849   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   ^cexp 12860   || cdvds 14983   Primecprime 15385   pCnt cpc 15541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542

This theorem is referenced by:  pczcl  15553  pcmul  15556  pcdiv  15557  pc1  15560  pczdvds  15567  pczndvds  15569  pczndvds2  15571  pcneg  15578