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Theorem ivthinclemlm 13170
Description: Lemma for ivthinc 13179. 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 7940 . . 3  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . 4  |-  ( ph  ->  B  e.  RR )
43rexrd 7940 . . 3  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . 4  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 8009 . . 3  |-  ( ph  ->  A  <_  B )
7 lbicc2 9912 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
82, 4, 6, 7syl3anc 1227 . 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 5481 . . . . 5  |-  ( w  =  A  ->  ( F `  w )  =  ( F `  A ) )
1211breq1d 3987 . . . 4  |-  ( w  =  A  ->  (
( F `  w
)  <  U  <->  ( F `  A )  <  U
) )
13 ivthinclem.l . . . 4  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
1412, 13elrab2 2881 . . 3  |-  ( A  e.  L  <->  ( A  e.  ( A [,] B
)  /\  ( F `  A )  <  U
) )
158, 10, 14sylanbrc 414 . 2  |-  ( ph  ->  A  e.  L )
16 eleq1 2227 . . 3  |-  ( q  =  A  ->  (
q  e.  L  <->  A  e.  L ) )
1716rspcev 2826 . 2  |-  ( ( A  e.  ( A [,] B )  /\  A  e.  L )  ->  E. q  e.  ( A [,] B ) q  e.  L )
188, 15, 17syl2anc 409 1  |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1342    e. wcel 2135   E.wrex 2443   {crab 2446    C_ wss 3112   class class class wbr 3977   ` cfv 5183  (class class class)co 5837   CCcc 7743   RRcr 7744   RR*cxr 7924    < clt 7925    <_ cle 7926   [,]cicc 9819   -cn->ccncf 13115
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-pre-ltirr 7857  ax-pre-lttrn 7859
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-iota 5148  df-fun 5185  df-fv 5191  df-ov 5840  df-oprab 5841  df-mpo 5842  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-icc 9823
This theorem is referenced by:  ivthinclemex  13178
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