Theorem pcval 14452

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 459	. . . . . 6 |- ( ( p = P /\ r = N ) -> r = N )
2	1	eqeq1d 2456	. . . . 5 |- ( ( p = P /\ r = N ) -> ( r = 0 <-> N = 0 ) )
3		eqeq1 2458	. . . . . . . 8 |- ( r = N -> ( r = ( x / y ) <-> N = ( x / y ) ) )
4		oveq1 6277	. . . . . . . . . . . . . 14 |- ( p = P -> ( p ^ n ) = ( P ^ n ) )
5	4	breq1d 4449	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || x <-> ( P ^ n ) || x ) )
6	5	rabbidv 3098	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x } )
7	6	supeq1d 7897	. . . . . . . . . . 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 2513	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = S )
10	4	breq1d 4449	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || y <-> ( P ^ n ) || y ) )
11	10	rabbidv 3098	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y } )
12	11	supeq1d 7897	. . . . . . . . . . 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 2513	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = T )
15	9, 14	oveq12d 6288	. . . . . . . . 9 |- ( p = P -> ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) = ( S - T ) )
16	15	eqeq2d 2468	. . . . . . . 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 872	. . . . . . 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 2972	. . . . . 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 5555	. . . . 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 3954	. . . 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 14445	. . . 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 11325	. . . . 5 |- +oo e. _V
23		iotaex 5551	. . . . 5 |- ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) e. _V
24	22, 23	ifex 3997	. . . 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 6406	. . 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 3941	. . 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 2515	. 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 645	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 367   = wceq 1398   e. wcel 1823   =/= wne 2649   E.wrex 2805   {crab 2808   ifcif 3929   class class class wbr 4439   iotacio 5532  (class class class)co 6270   supcsup 7892   RRcr 9480   0cc0 9481   +oocpnf 9614   < clt 9617   - cmin 9796   / cdiv 10202   NNcn 10531   NN0cn0 10791   ZZcz 10860   QQcq 11183   ^cexp 12148   || cdvds 14070   Primecprime 14301   pCnt cpc 14444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-sup 7893  df-pnf 9619  df-xr 9621  df-pc 14445

This theorem is referenced by:  pczpre  14455  pcdiv  14460