Theorem pcval 14806

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 463	. . . . . 6 |- ( ( p = P /\ r = N ) -> r = N )
2	1	eqeq1d 2455	. . . . 5 |- ( ( p = P /\ r = N ) -> ( r = 0 <-> N = 0 ) )
3		eqeq1 2457	. . . . . . . 8 |- ( r = N -> ( r = ( x / y ) <-> N = ( x / y ) ) )
4		oveq1 6302	. . . . . . . . . . . . . 14 |- ( p = P -> ( p ^ n ) = ( P ^ n ) )
5	4	breq1d 4415	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || x <-> ( P ^ n ) || x ) )
6	5	rabbidv 3038	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x } )
7	6	supeq1d 7965	. . . . . . . . . . 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 2505	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = S )
10	4	breq1d 4415	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || y <-> ( P ^ n ) || y ) )
11	10	rabbidv 3038	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y } )
12	11	supeq1d 7965	. . . . . . . . . . 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 2505	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = T )
15	9, 14	oveq12d 6313	. . . . . . . . 9 |- ( p = P -> ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) = ( S - T ) )
16	15	eqeq2d 2463	. . . . . . . 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 886	. . . . . . 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 2910	. . . . . 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 5570	. . . . 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 3908	. . . 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 14799	. . . 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 11420	. . . . 5 |- +oo e. _V
23		iotaex 5566	. . . . 5 |- ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) e. _V
24	22, 23	ifex 3951	. . . 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 6432	. . 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 3895	. . 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 2507	. 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 653	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 371   = wceq 1446   e. wcel 1889   =/= wne 2624   E.wrex 2740   {crab 2743   ifcif 3883   class class class wbr 4405   iotacio 5547  (class class class)co 6295   supcsup 7959   RRcr 9543   0cc0 9544   +oocpnf 9677   < clt 9680   - cmin 9865   / cdiv 10276   NNcn 10616   NN0cn0 10876   ZZcz 10944   QQcq 11271   ^cexp 12279   || cdvds 14317   Primecprime 14634   pCnt cpc 14798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5549  df-fun 5587  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-sup 7961  df-pnf 9682  df-xr 9684  df-pc 14799

This theorem is referenced by:  pczpre  14809  pcdiv  14814