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Theorem ivthinclemum 14274
Description: Lemma for ivthinc 14282. The upper 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
ivthinclemum  |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  R )
Distinct variable groups:    A, r    w, A    B, r    w, B   
w, F    R, r    w, U
Allowed substitution hints:    ph( x, y, w, r)    A( x, y)    B( x, y)    D( x, y, w, r)    R( x, y, w)    U( x, y, r)    F( x, y, r)    L( x, y, w, r)

Proof of Theorem ivthinclemum
StepHypRef Expression
1 ivth.1 . . . 4  |-  ( ph  ->  A  e.  RR )
21rexrd 8010 . . 3  |-  ( ph  ->  A  e.  RR* )
3 ivth.2 . . . 4  |-  ( ph  ->  B  e.  RR )
43rexrd 8010 . . 3  |-  ( ph  ->  B  e.  RR* )
5 ivth.4 . . . 4  |-  ( ph  ->  A  <  B )
61, 3, 5ltled 8079 . . 3  |-  ( ph  ->  A  <_  B )
7 ubicc2 9988 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
82, 4, 6, 7syl3anc 1238 . 2  |-  ( ph  ->  B  e.  ( A [,] B ) )
9 ivth.9 . . . 4  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
109simprd 114 . . 3  |-  ( ph  ->  U  <  ( F `
 B ) )
11 fveq2 5517 . . . . 5  |-  ( w  =  B  ->  ( F `  w )  =  ( F `  B ) )
1211breq2d 4017 . . . 4  |-  ( w  =  B  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  B ) ) )
13 ivthinclem.r . . . 4  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
1412, 13elrab2 2898 . . 3  |-  ( B  e.  R  <->  ( B  e.  ( A [,] B
)  /\  U  <  ( F `  B ) ) )
158, 10, 14sylanbrc 417 . 2  |-  ( ph  ->  B  e.  R )
16 eleq1 2240 . . 3  |-  ( r  =  B  ->  (
r  e.  R  <->  B  e.  R ) )
1716rspcev 2843 . 2  |-  ( ( B  e.  ( A [,] B )  /\  B  e.  R )  ->  E. r  e.  ( A [,] B ) r  e.  R )
188, 15, 17syl2anc 411 1  |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   E.wrex 2456   {crab 2459    C_ wss 3131   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   RR*cxr 7994    < clt 7995    <_ cle 7996   [,]cicc 9894   -cn->ccncf 14218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-pre-ltirr 7926  ax-pre-lttrn 7928
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-icc 9898
This theorem is referenced by:  ivthinclemex  14281
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