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Theorem ovolicc2lem1 18892
Description: Lemma for ovolicc2 18897. (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 )
2 ffvelrn 5679 . . . . . 6  |-  ( ( G : U --> NN  /\  X  e.  U )  ->  ( G `  X
)  e.  NN )
31, 2sylan 457 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( G `  X )  e.  NN )
4 ovolicc2.5 . . . . . . 7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
5 inss2 3403 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
6 fss 5413 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR ) )  ->  F : NN --> ( RR  X.  RR ) )
74, 5, 6sylancl 643 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
8 fvco3 5612 . . . . . 6  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  X )  e.  NN )  -> 
( ( (,)  o.  F ) `  ( G `  X )
)  =  ( (,) `  ( F `  ( G `  X )
) ) )
97, 8sylan 457 . . . . 5  |-  ( (
ph  /\  ( G `  X )  e.  NN )  ->  ( ( (,) 
o.  F ) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X )
) ) )
103, 9syldan 456 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X ) ) ) )
11 ovolicc2.9 . . . . . 6  |-  ( (
ph  /\  t  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  t )
1211ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t )
13 fveq2 5541 . . . . . . . 8  |-  ( t  =  X  ->  ( G `  t )  =  ( G `  X ) )
1413fveq2d 5545 . . . . . . 7  |-  ( t  =  X  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  ( ( (,)  o.  F ) `  ( G `  X )
) )
15 id 19 . . . . . . 7  |-  ( t  =  X  ->  t  =  X )
1614, 15eqeq12d 2310 . . . . . 6  |-  ( t  =  X  ->  (
( ( (,)  o.  F ) `  ( G `  t )
)  =  t  <->  ( ( (,)  o.  F ) `  ( G `  X ) )  =  X ) )
1716rspccva 2896 . . . . 5  |-  ( ( A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t  /\  X  e.  U )  ->  ( ( (,)  o.  F ) `  ( G `  X )
)  =  X )
1812, 17sylan 457 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  X )
197adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  F : NN --> ( RR  X.  RR ) )
20 ffvelrn 5679 . . . . . . . 8  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  X )  e.  NN )  -> 
( F `  ( G `  X )
)  e.  ( RR 
X.  RR ) )
2119, 3, 20syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  e.  ( RR  X.  RR ) )
22 1st2nd2 6175 . . . . . . 7  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( F `
 ( G `  X ) )  = 
<. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2321, 22syl 15 . . . . . 6  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  = 
<. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2423fveq2d 5545 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `
 ( G `  X ) ) )
>. ) )
25 df-ov 5877 . . . . 5  |-  ( ( 1st `  ( F `
 ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2624, 25syl6eqr 2346 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2710, 18, 263eqtr3d 2336 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  ( ( 1st `  ( F `  ( G `  X )
) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2827eleq2d 2363 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
29 xp1st 6165 . . . 4  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( 1st `  ( F `  ( G `  X )
) )  e.  RR )
3021, 29syl 15 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR )
31 xp2nd 6166 . . . 4  |-  ( ( F `  ( G `
 X ) )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( F `  ( G `  X )
) )  e.  RR )
3221, 31syl 15 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR )
33 rexr 8893 . . . 4  |-  ( ( 1st `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR* )
34 rexr 8893 . . . 4  |-  ( ( 2nd `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )
35 elioo2 10713 . . . 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 )
) ) ) ) )
3633, 34, 35syl2an 463 . . 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 )
) ) ) ) )
3730, 32, 36syl2anc 642 . 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 )
) ) ) ) )
3828, 37bitrd 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   <.cop 3656   U.cuni 3843   class class class wbr 4039    X. cxp 4703   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Fincfn 6879   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   (,)cioo 10672   [,]cicc 10675    seq cseq 11062   abscabs 11735
This theorem is referenced by:  ovolicc2lem2  18893  ovolicc2lem3  18894  ovolicc2lem4  18895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioo 10676
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