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Theorem ovolicc2lem1 22094
Description: Lemma for ovolicc2 22099. (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 6007 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( G `  X )  e.  NN )
3 ovolicc2.5 . . . . . . 7  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3705 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 fss 5721 . . . . . . 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 660 . . . . . 6  |-  ( ph  ->  F : NN --> ( RR 
X.  RR ) )
7 fvco3 5925 . . . . . 6  |-  ( ( F : NN --> ( RR 
X.  RR )  /\  ( G `  X )  e.  NN )  -> 
( ( (,)  o.  F ) `  ( G `  X )
)  =  ( (,) `  ( F `  ( G `  X )
) ) )
86, 7sylan 469 . . . . 5  |-  ( (
ph  /\  ( G `  X )  e.  NN )  ->  ( ( (,) 
o.  F ) `  ( G `  X ) )  =  ( (,) `  ( F `  ( G `  X )
) ) )
92, 8syldan 468 . . . 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 2868 . . . . 5  |-  ( ph  ->  A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t )
12 fveq2 5848 . . . . . . . 8  |-  ( t  =  X  ->  ( G `  t )  =  ( G `  X ) )
1312fveq2d 5852 . . . . . . 7  |-  ( t  =  X  ->  (
( (,)  o.  F
) `  ( G `  t ) )  =  ( ( (,)  o.  F ) `  ( G `  X )
) )
14 id 22 . . . . . . 7  |-  ( t  =  X  ->  t  =  X )
1513, 14eqeq12d 2476 . . . . . 6  |-  ( t  =  X  ->  (
( ( (,)  o.  F ) `  ( G `  t )
)  =  t  <->  ( ( (,)  o.  F ) `  ( G `  X ) )  =  X ) )
1615rspccva 3206 . . . . 5  |-  ( ( A. t  e.  U  ( ( (,)  o.  F ) `  ( G `  t )
)  =  t  /\  X  e.  U )  ->  ( ( (,)  o.  F ) `  ( G `  X )
)  =  X )
1711, 16sylan 469 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  (
( (,)  o.  F
) `  ( G `  X ) )  =  X )
186adantr 463 . . . . . . . 8  |-  ( (
ph  /\  X  e.  U )  ->  F : NN --> ( RR  X.  RR ) )
1918, 2ffvelrnd 6008 . . . . . . 7  |-  ( (
ph  /\  X  e.  U )  ->  ( F `  ( G `  X ) )  e.  ( RR  X.  RR ) )
20 1st2nd2 6810 . . . . . . 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 5852 . . . . 5  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `
 ( G `  X ) ) )
>. ) )
23 df-ov 6273 . . . . 5  |-  ( ( 1st `  ( F `
 ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) )  =  ( (,) `  <. ( 1st `  ( F `  ( G `  X ) ) ) ,  ( 2nd `  ( F `  ( G `  X ) ) )
>. )
2422, 23syl6eqr 2513 . . . 4  |-  ( (
ph  /\  X  e.  U )  ->  ( (,) `  ( F `  ( G `  X ) ) )  =  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
259, 17, 243eqtr3d 2503 . . 3  |-  ( (
ph  /\  X  e.  U )  ->  X  =  ( ( 1st `  ( F `  ( G `  X )
) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) )
2625eleq2d 2524 . 2  |-  ( (
ph  /\  X  e.  U )  ->  ( P  e.  X  <->  P  e.  ( ( 1st `  ( F `  ( G `  X ) ) ) (,) ( 2nd `  ( F `  ( G `  X ) ) ) ) ) )
27 xp1st 6803 . . . 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 6804 . . . 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 9628 . . . 4  |-  ( ( 1st `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 1st `  ( F `  ( G `  X ) ) )  e.  RR* )
32 rexr 9628 . . . 4  |-  ( ( 2nd `  ( F `
 ( G `  X ) ) )  e.  RR  ->  ( 2nd `  ( F `  ( G `  X ) ) )  e.  RR* )
33 elioo2 11573 . . . 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 475 . . 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 659 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   <.cop 4022   U.cuni 4235   class class class wbr 4439    X. cxp 4986   ran crn 4989    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Fincfn 7509   RRcr 9480   1c1 9482    + caddc 9484   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   (,)cioo 11532   [,]cicc 11535    seqcseq 12089   abscabs 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioo 11536
This theorem is referenced by:  ovolicc2lem2  22095  ovolicc2lem3  22096  ovolicc2lem4  22097
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