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Theorem ivthinclemlm 12770
Description: Lemma for ivthinc 12779. The lower cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.)
Hypotheses
Ref Expression
ivth.1  |-  ( ph  ->  A  e.  RR )
ivth.2  |-  ( ph  ->  B  e.  RR )
ivth.3  |-  ( ph  ->  U  e.  RR )
ivth.4  |-  ( ph  ->  A  <  B )
ivth.5  |-  ( ph  ->  ( A [,] B
)  C_  D )
ivth.7  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
ivth.8  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
ivth.9  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
ivthinc.i  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
ivthinclem.l  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
ivthinclem.r  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
Assertion
Ref Expression
ivthinclemlm  |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
Distinct variable groups:    A, q    w, A    B, q    w, B   
w, F    L, q    w, U
Allowed substitution hints:    ph( x, y, w, q)    A( x, y)    B( x, y)    D( x, y, w, q)    R( x, y, w, q)    U( x, y, q)    F( x, y, q)    L( x, y, w)

Proof of Theorem ivthinclemlm
StepHypRef Expression
1 ivth.1 . . . 4  |-  ( ph  ->  A  e.  RR )
21rexrd 7808 . . 3  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . 4  |-  ( ph  ->  B  e.  RR )
43rexrd 7808 . . 3  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . 4  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 7874 . . 3  |-  ( ph  ->  A  <_  B )
7 lbicc2 9760 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
82, 4, 6, 7syl3anc 1216 . 2  |-  ( ph  ->  A  e.  ( A [,] B ) )
9 ivth.9 . . . 4  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
109simpld 111 . . 3  |-  ( ph  ->  ( F `  A
)  <  U )
11 fveq2 5414 . . . . 5  |-  ( w  =  A  ->  ( F `  w )  =  ( F `  A ) )
1211breq1d 3934 . . . 4  |-  ( w  =  A  ->  (
( F `  w
)  <  U  <->  ( F `  A )  <  U
) )
13 ivthinclem.l . . . 4  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
1412, 13elrab2 2838 . . 3  |-  ( A  e.  L  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <  U
) )
158, 10, 14sylanbrc 413 . 2  |-  ( ph  ->  A  e.  L )
16 eleq1 2200 . . 3  |-  ( q  =  A  ->  (
q  e.  L  <->  A  e.  L ) )
1716rspcev 2784 . 2  |-  ( ( A  e.  ( A [,] B )  /\  A  e.  L )  ->  E. q  e.  ( A [,] B ) q  e.  L )
188, 15, 17syl2anc 408 1  |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   E.wrex 2415   {crab 2418    C_ wss 3066   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   RR*cxr 7792    < clt 7793    <_ cle 7794   [,]cicc 9667   -cn->ccncf 12715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725  ax-pre-lttrn 7727
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-icc 9671
This theorem is referenced by:  ivthinclemex  12778
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