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Theorem ovolicc2lem1 21000
Description: Lemma for ovolicc2 21005. (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 5843 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( G `  X )  e.  NN )
3 ovolicc2.5 . . . . . . 7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3571 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5567 . . . . . . 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 5768 . . . . . 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 2799 . . . . 5  |-  ( ph  ->  A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t )
12 fveq2 5691 . . . . . . . 8  |-  ( t  =  X  ->  ( G `  t )  =  ( G `  X ) )
1312fveq2d 5695 . . . . . . 7  |-  ( t  =  X  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  ( ( (,)  o.  F ) `  ( G `  X )
) )
14 id 22 . . . . . . 7  |-  ( t  =  X  ->  t  =  X )
1513, 14eqeq12d 2457 . . . . . 6  |-  ( t  =  X  ->  (
( ( (,)  o.  F ) `  ( G `  t )
)  =  t  <->  ( ( (,)  o.  F ) `  ( G `  X ) )  =  X ) )
1615rspccva 3072 . . . . 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 5844 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  e.  ( RR  X.  RR ) )
20 1st2nd2 6613 . . . . . . 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 5695 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `
 ( G `  X ) ) )
>. ) )
23 df-ov 6094 . . . . 5  |-  ( ( 1st `  ( F `
 ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2422, 23syl6eqr 2493 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
259, 17, 243eqtr3d 2483 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  ( ( 1st `  ( F `  ( G `  X )
) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2625eleq2d 2510 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
27 xp1st 6606 . . . 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 6607 . . . 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 9429 . . . 4  |-  ( ( 1st `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR* )
32 rexr 9429 . . . 4  |-  ( ( 2nd `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )
33 elioo2 11341 . . . 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 965    = wceq 1369    e. wcel 1756   A.wral 2715    i^i cin 3327    C_ wss 3328   ~Pcpw 3860   <.cop 3883   U.cuni 4091   class class class wbr 4292    X. cxp 4838   ran crn 4841    o. ccom 4844   -->wf 5414   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   Fincfn 7310   RRcr 9281   1c1 9283    + caddc 9285   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595   NNcn 10322   (,)cioo 11300   [,]cicc 11303    seqcseq 11806   abscabs 12723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-ioo 11304
This theorem is referenced by:  ovolicc2lem2  21001  ovolicc2lem3  21002  ovolicc2lem4  21003
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