Theorem pcval 15549

Description: The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Hypotheses

Ref	Expression
pcval.1	|- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
pcval.2	|- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )

Assertion

Ref	Expression
pcval	|- ( ( 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 = ( S - T ) ) ) )

Distinct variable groups:   x, n, y, z, N   P, n, x, y, z   z, S   z, T

Allowed substitution hints:   S( x, y, n)   T( x, y, n)

Proof of Theorem pcval

Dummy variables p r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( p = P /\ r = N ) -> r = N )
2	1	eqeq1d 2624	. . . . 5 |- ( ( p = P /\ r = N ) -> ( r = 0 <-> N = 0 ) )
3		eqeq1 2626	. . . . . . . 8 |- ( r = N -> ( r = ( x / y ) <-> N = ( x / y ) ) )
4		oveq1 6657	. . . . . . . . . . . . . 14 |- ( p = P -> ( p ^ n ) = ( P ^ n ) )
5	4	breq1d 4663	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || x <-> ( P ^ n ) || x ) )
6	5	rabbidv 3189	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x } )
7	6	supeq1d 8352	. . . . . . . . . . 11 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) )
8		pcval.1	. . . . . . . . . . 11 |- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
9	7, 8	syl6eqr 2674	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = S )
10	4	breq1d 4663	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || y <-> ( P ^ n ) || y ) )
11	10	rabbidv 3189	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y } )
12	11	supeq1d 8352	. . . . . . . . . . 11 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) )
13		pcval.2	. . . . . . . . . . 11 |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
14	12, 13	syl6eqr 2674	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = T )
15	9, 14	oveq12d 6668	. . . . . . . . 9 |- ( p = P -> ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) = ( S - T ) )
16	15	eqeq2d 2632	. . . . . . . 8 |- ( p = P -> ( z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) <-> z = ( S - T ) ) )
17	3, 16	bi2anan9r 918	. . . . . . 7 |- ( ( p = P /\ r = N ) -> ( ( r = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) ) <-> ( N = ( x / y ) /\ z = ( S - T ) ) ) )
18	17	2rexbidv 3057	. . . . . 6 |- ( ( p = P /\ r = N ) -> ( E. x e. ZZ E. y e. NN ( r = ( 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 ) /\ z = ( S - T ) ) ) )
19	18	iotabidv 5872	. . . . 5 |- ( ( p = P /\ r = N ) -> ( iota z E. x e. ZZ E. y e. NN ( r = ( x / y ) /\ z = ( 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 = ( S - T ) ) ) )
20	2, 19	ifbieq2d 4111	. . . 4 |- ( ( p = P /\ r = N ) -> if ( r = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( r = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) ) ) ) = if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) )
21		df-pc 15542	. . . 4 |- pCnt = ( p e. Prime , r e. QQ |-> if ( r = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( r = ( x / y ) /\ z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) ) ) ) )
22		pnfex 10093	. . . . 5 |- +oo e. _V
23		iotaex 5868	. . . . 5 |- ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) e. _V
24	22, 23	ifex 4156	. . . 4 |- if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) e. _V
25	20, 21, 24	ovmpt2a 6791	. . 3 |- ( ( P e. Prime /\ N e. QQ ) -> ( P pCnt N ) = if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) )
26		ifnefalse 4098	. . 3 |- ( N =/= 0 -> if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) = ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) )
27	25, 26	sylan9eq 2676	. 2 |- ( ( ( 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 = ( S - T ) ) ) )
28	27	anasss 679	1 |- ( ( 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 = ( S - T ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   ifcif 4086   class class class wbr 4653   iotacio 5849  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   +oocpnf 10071   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sup 8348  df-pnf 10076  df-xr 10078  df-pc 15542

This theorem is referenced by:  pczpre  15552  pcdiv  15557