MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolicc2lem1 Structured version   Unicode version

Theorem ovolicc2lem1 21660
Description: Lemma for ovolicc2 21665. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ovolicc.1  |-  ( ph  ->  A  e.  RR )
ovolicc.2  |-  ( ph  ->  B  e.  RR )
ovolicc.3  |-  ( ph  ->  A  <_  B )
ovolicc2.4  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
ovolicc2.5  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
ovolicc2.6  |-  ( ph  ->  U  e.  ( ~P
ran  ( (,)  o.  F )  i^i  Fin ) )
ovolicc2.7  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
ovolicc2.8  |-  ( ph  ->  G : U --> NN )
ovolicc2.9  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
Assertion
Ref Expression
ovolicc2lem1  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X )
) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X )
) ) ) ) )
Distinct variable groups:    t, A    t, B    t, F    t, G    ph, t    t, U   
t, X
Allowed substitution hints:    P( t)    S( t)

Proof of Theorem ovolicc2lem1
StepHypRef Expression
1 ovolicc2.8 . . . . . 6  |-  ( ph  ->  G : U --> NN )
21ffvelrnda 6019 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( G `  X )  e.  NN )
3 ovolicc2.5 . . . . . . 7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3719 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5737 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR ) )  ->  F : NN --> ( RR  X.  RR ) )
63, 4, 5sylancl 662 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
7 fvco3 5942 . . . . . 6  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  X )  e.  NN )  -> 
( ( (,)  o.  F ) `  ( G `  X )
)  =  ( (,) `  ( F `  ( G `  X )
) ) )
86, 7sylan 471 . . . . 5  |-  ( (
ph  /\  ( G `  X )  e.  NN )  ->  ( ( (,) 
o.  F ) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X )
) ) )
92, 8syldan 470 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X ) ) ) )
10 ovolicc2.9 . . . . . 6  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
1110ralrimiva 2878 . . . . 5  |-  ( ph  ->  A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t )
12 fveq2 5864 . . . . . . . 8  |-  ( t  =  X  ->  ( G `  t )  =  ( G `  X ) )
1312fveq2d 5868 . . . . . . 7  |-  ( t  =  X  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  ( ( (,)  o.  F ) `  ( G `  X )
) )
14 id 22 . . . . . . 7  |-  ( t  =  X  ->  t  =  X )
1513, 14eqeq12d 2489 . . . . . 6  |-  ( t  =  X  ->  (
( ( (,)  o.  F ) `  ( G `  t )
)  =  t  <->  ( ( (,)  o.  F ) `  ( G `  X ) )  =  X ) )
1615rspccva 3213 . . . . 5  |-  ( ( A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t  /\  X  e.  U )  ->  ( ( (,)  o.  F ) `  ( G `  X )
)  =  X )
1711, 16sylan 471 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  X )
186adantr 465 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  F : NN --> ( RR  X.  RR ) )
1918, 2ffvelrnd 6020 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  e.  ( RR  X.  RR ) )
20 1st2nd2 6818 . . . . . . 7  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( F `
 ( G `  X ) )  = 
<. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2119, 20syl 16 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  = 
<. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2221fveq2d 5868 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `
 ( G `  X ) ) )
>. ) )
23 df-ov 6285 . . . . 5  |-  ( ( 1st `  ( F `
 ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2422, 23syl6eqr 2526 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
259, 17, 243eqtr3d 2516 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  ( ( 1st `  ( F `  ( G `  X )
) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2625eleq2d 2537 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
27 xp1st 6811 . . . 4  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  ( G `  X )
) )  e.  RR )
2819, 27syl 16 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR )
29 xp2nd 6812 . . . 4  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  ( G `  X )
) )  e.  RR )
3019, 29syl 16 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR )
31 rexr 9635 . . . 4  |-  ( ( 1st `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR* )
32 rexr 9635 . . . 4  |-  ( ( 2nd `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )
33 elioo2 11566 . . . 4  |-  ( ( ( 1st `  ( F `  ( G `  X ) ) )  e.  RR*  /\  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )  ->  ( P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X )
) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X )
) ) ) ) )
3431, 32, 33syl2an 477 . . 3  |-  ( ( ( 1st `  ( F `  ( G `  X ) ) )  e.  RR  /\  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR )  ->  ( P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X )
) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X )
) ) ) ) )
3528, 30, 34syl2anc 661 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X )
) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X )
) ) ) ) )
3626, 35bitrd 253 1  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  ( P  e.  RR  /\  ( 1st `  ( F `  ( G `  X )
) )  <  P  /\  P  <  ( 2nd `  ( F `  ( G `  X )
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   <.cop 4033   U.cuni 4245   class class class wbr 4447    X. cxp 4997   ran crn 5000    o. ccom 5003   -->wf 5582   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Fincfn 7513   RRcr 9487   1c1 9489    + caddc 9491   RR*cxr 9623    < clt 9624    <_ cle 9625    - cmin 9801   NNcn 10532   (,)cioo 11525   [,]cicc 11528    seqcseq 12070   abscabs 13024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-ioo 11529
This theorem is referenced by:  ovolicc2lem2  21661  ovolicc2lem3  21662  ovolicc2lem4  21663
  Copyright terms: Public domain W3C validator