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Theorem ovolicc2lem1 19413
Description: Lemma for ovolicc2 19418. (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 5870 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( G `  X )  e.  NN )
3 ovolicc2.5 . . . . . . 7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3562 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5599 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
7 fvco3 5800 . . . . . 6  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  X )  e.  NN )  -> 
( ( (,)  o.  F ) `  ( G `  X )
)  =  ( (,) `  ( F `  ( G `  X )
) ) )
86, 7sylan 458 . . . . 5  |-  ( (
ph  /\  ( G `  X )  e.  NN )  ->  ( ( (,) 
o.  F ) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X )
) ) )
92, 8syldan 457 . . . 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 2789 . . . . 5  |-  ( ph  ->  A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t )
12 fveq2 5728 . . . . . . . 8  |-  ( t  =  X  ->  ( G `  t )  =  ( G `  X ) )
1312fveq2d 5732 . . . . . . 7  |-  ( t  =  X  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  ( ( (,)  o.  F ) `  ( G `  X )
) )
14 id 20 . . . . . . 7  |-  ( t  =  X  ->  t  =  X )
1513, 14eqeq12d 2450 . . . . . 6  |-  ( t  =  X  ->  (
( ( (,)  o.  F ) `  ( G `  t )
)  =  t  <->  ( ( (,)  o.  F ) `  ( G `  X ) )  =  X ) )
1615rspccva 3051 . . . . 5  |-  ( ( A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t  /\  X  e.  U )  ->  ( ( (,)  o.  F ) `  ( G `  X )
)  =  X )
1711, 16sylan 458 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  X )
186adantr 452 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  F : NN --> ( RR  X.  RR ) )
1918, 2ffvelrnd 5871 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  e.  ( RR  X.  RR ) )
20 1st2nd2 6386 . . . . . . 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 5732 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `
 ( G `  X ) ) )
>. ) )
23 df-ov 6084 . . . . 5  |-  ( ( 1st `  ( F `
 ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2422, 23syl6eqr 2486 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
259, 17, 243eqtr3d 2476 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  ( ( 1st `  ( F `  ( G `  X )
) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2625eleq2d 2503 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
27 xp1st 6376 . . . 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 6377 . . . 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 9130 . . . 4  |-  ( ( 1st `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR* )
32 rexr 9130 . . . 4  |-  ( ( 2nd `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )
33 elioo2 10957 . . . 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 464 . . 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 643 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   <.cop 3817   U.cuni 4015   class class class wbr 4212    X. cxp 4876   ran crn 4879    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Fincfn 7109   RRcr 8989   1c1 8991    + caddc 8993   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   (,)cioo 10916   [,]cicc 10919    seq cseq 11323   abscabs 12039
This theorem is referenced by:  ovolicc2lem2  19414  ovolicc2lem3  19415  ovolicc2lem4  19416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-ioo 10920
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