Theorem pcval 13909

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 461	. . . . . 6 |- ( ( p = P /\ r = N ) -> r = N )
2	1	eqeq1d 2449	. . . . 5 |- ( ( p = P /\ r = N ) -> ( r = 0 <-> N = 0 ) )
3		eqeq1 2447	. . . . . . . 8 |- ( r = N -> ( r = ( x / y ) <-> N = ( x / y ) ) )
4		oveq1 6096	. . . . . . . . . . . . . 14 |- ( p = P -> ( p ^ n ) = ( P ^ n ) )
5	4	breq1d 4300	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || x <-> ( P ^ n ) || x ) )
6	5	rabbidv 2962	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x } )
7	6	supeq1d 7694	. . . . . . . . . . 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 2491	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = S )
10	4	breq1d 4300	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || y <-> ( P ^ n ) || y ) )
11	10	rabbidv 2962	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y } )
12	11	supeq1d 7694	. . . . . . . . . . 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 2491	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = T )
15	9, 14	oveq12d 6107	. . . . . . . . 9 |- ( p = P -> ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) = ( S - T ) )
16	15	eqeq2d 2452	. . . . . . . 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 869	. . . . . . 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 2756	. . . . . 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 5400	. . . . 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 3812	. . . 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 13902	. . . 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 11091	. . . . 5 |- +oo e. _V
23		iotaex 5396	. . . . 5 |- ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) e. _V
24	22, 23	ifex 3856	. . . 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 6219	. . 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 3799	. . 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 2493	. 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 647	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 369   = wceq 1369   e. wcel 1756   =/= wne 2604   E.wrex 2714   {crab 2717   ifcif 3789   class class class wbr 4290   iotacio 5377  (class class class)co 6089   supcsup 7688   RRcr 9279   0cc0 9280   +oocpnf 9413   < clt 9416   - cmin 9593   / cdiv 9991   NNcn 10320   NN0cn0 10577   ZZcz 10644   QQcq 10951   ^cexp 11863   || cdivides 13533   Primecprime 13761   pCnt cpc 13901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-sup 7689  df-pnf 9418  df-xr 9420  df-pc 13902

This theorem is referenced by:  pczpre  13912  pcdiv  13917