Theorem pcval 13019

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		simpl 109	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> P e. Prime )
2		simprl 531	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> N e. QQ )
3		ifnefalse 3637	. . . . 5 |- ( 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 ) ) ) )
4	3	ad2antll 491	. . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ 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 ) ) ) )
5		pcval.1	. . . . . 6 |- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
6		pcval.2	. . . . . 6 |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
7	5, 6	pceu 13018	. . . . 5 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> E! z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) )
8		euiotaex 5334	. . . . 5 |- ( E! 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 ) ) ) e. _V )
9	7, 8	syl 14	. . . 4 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) e. _V )
10	4, 9	eqeltrd 2311	. . 3 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) e. _V )
11		simpr 110	. . . . . 6 |- ( ( p = P /\ r = N ) -> r = N )
12	11	eqeq1d 2243	. . . . 5 |- ( ( p = P /\ r = N ) -> ( r = 0 <-> N = 0 ) )
13		eqeq1 2241	. . . . . . . 8 |- ( r = N -> ( r = ( x / y ) <-> N = ( x / y ) ) )
14		oveq1 6065	. . . . . . . . . . . . . 14 |- ( p = P -> ( p ^ n ) = ( P ^ n ) )
15	14	breq1d 4124	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || x <-> ( P ^ n ) || x ) )
16	15	rabbidv 2804	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x } )
17	16	supeq1d 7291	. . . . . . . . . . 11 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < ) )
18	17, 5	eqtr4di 2285	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) = S )
19	14	breq1d 4124	. . . . . . . . . . . . 13 |- ( p = P -> ( ( p ^ n ) || y <-> ( P ^ n ) || y ) )
20	19	rabbidv 2804	. . . . . . . . . . . 12 |- ( p = P -> { n e. NN0 | ( p ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y } )
21	20	supeq1d 7291	. . . . . . . . . . 11 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < ) )
22	21, 6	eqtr4di 2285	. . . . . . . . . 10 |- ( p = P -> sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) = T )
23	18, 22	oveq12d 6076	. . . . . . . . 9 |- ( p = P -> ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) = ( S - T ) )
24	23	eqeq2d 2246	. . . . . . . 8 |- ( p = P -> ( z = ( sup ( { n e. NN0 | ( p ^ n ) || x } , RR , < ) - sup ( { n e. NN0 | ( p ^ n ) || y } , RR , < ) ) <-> z = ( S - T ) ) )
25	13, 24	bi2anan9r 611	. . . . . . 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 ) ) ) )
26	25	2rexbidv 2569	. . . . . 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 ) ) ) )
27	26	iotabidv 5340	. . . . 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 ) ) ) )
28	12, 27	ifbieq2d 3651	. . . 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 ) ) ) ) )
29		df-pc 13008	. . . 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 , < ) ) ) ) ) )
30	28, 29	ovmpoga 6191	. . 3 |- ( ( P e. Prime /\ N e. QQ /\ if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) e. _V ) -> ( P pCnt N ) = if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) )
31	1, 2, 10, 30	syl3anc 1274	. 2 |- ( ( P e. Prime /\ ( N e. QQ /\ N =/= 0 ) ) -> ( P pCnt N ) = if ( N = 0 , +oo , ( iota z E. x e. ZZ E. y e. NN ( N = ( x / y ) /\ z = ( S - T ) ) ) ) )
32	31, 4	eqtrd 2267	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 104   = wceq 1398   E!weu 2082   e. wcel 2205   =/= wne 2414   E.wrex 2523   {crab 2526   _Vcvv 2815   ifcif 3624   class class class wbr 4114   iotacio 5315  (class class class)co 6058   supcsup 7286   RRcr 8142   0cc0 8143   +oocpnf 8321   < clt 8324   - cmin 8460   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   QQcq 9969   ^cexp 10924   || cdvds 12498   Primecprime 12829   pCnt cpc 13007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830  df-pc 13008

This theorem is referenced by:  pczpre  13020  pcdiv  13025