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Theorem ivthinclemlr 12773
Description: Lemma for ivthinc 12779. The lower cut is rounded. (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
ivthinclemlr  |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
Distinct variable groups:    A, r, w   
x, A, y, r    B, r, w    x, B, y    w, F    x, F, y    L, r, x, y    w, U    ph, q,
r, x, y    w, q
Allowed substitution hints:    ph( w)    A( q)    B( q)    D( x, y, w, r, q)    R( x, y, w, r, q)    U( x, y, r, q)    F( r, q)    L( w, q)

Proof of Theorem ivthinclemlr
StepHypRef Expression
1 ivth.1 . . . . . 6  |-  ( ph  ->  A  e.  RR )
21ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  e.  RR )
3 ivth.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
43ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  B  e.  RR )
5 ivth.3 . . . . . 6  |-  ( ph  ->  U  e.  RR )
65ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  U  e.  RR )
7 ivth.4 . . . . . 6  |-  ( ph  ->  A  <  B )
87ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  <  B )
9 ivth.5 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  D )
109ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  ( A [,] B )  C_  D )
11 ivth.7 . . . . . 6  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1211ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  F  e.  ( D -cn-> CC ) )
13 ivth.8 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
1413adantlr 468 . . . . . 6  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  x  e.  ( A [,] B
) )  ->  ( F `  x )  e.  RR )
1514adantlr 468 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  q  e.  L )  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
16 ivth.9 . . . . . 6  |-  ( ph  ->  ( ( F `  A )  <  U  /\  U  <  ( F `
 B ) ) )
1716ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  (
( F `  A
)  <  U  /\  U  <  ( F `  B ) ) )
18 ivthinc.i . . . . . . 7  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
1918adantllr 472 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  <  y
) )  ->  ( F `  x )  <  ( F `  y
) )
2019adantllr 472 . . . . 5  |-  ( ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  q  e.  L
)  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  <  y ) )  -> 
( F `  x
)  <  ( F `  y ) )
21 ivthinclem.l . . . . 5  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
22 ivthinclem.r . . . . 5  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
23 simpr 109 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  q  e.  L )
242, 4, 6, 8, 10, 12, 15, 17, 20, 21, 22, 23ivthinclemlopn 12772 . . . 4  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  E. r  e.  L  q  <  r )
2524ex 114 . . 3  |-  ( (
ph  /\  q  e.  ( A [,] B ) )  ->  ( q  e.  L  ->  E. r  e.  L  q  <  r ) )
26 simpllr 523 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  ( A [,] B ) )
27 fveq2 5414 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
2827eleq1d 2206 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  e.  RR  <->  ( F `  q )  e.  RR ) )
2913ralrimiva 2503 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3029ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3128, 30, 26rspcdva 2789 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  e.  RR )
32 fveq2 5414 . . . . . . . 8  |-  ( x  =  r  ->  ( F `  x )  =  ( F `  r ) )
3332eleq1d 2206 . . . . . . 7  |-  ( x  =  r  ->  (
( F `  x
)  e.  RR  <->  ( F `  r )  e.  RR ) )
34 fveq2 5414 . . . . . . . . . . 11  |-  ( w  =  r  ->  ( F `  w )  =  ( F `  r ) )
3534breq1d 3934 . . . . . . . . . 10  |-  ( w  =  r  ->  (
( F `  w
)  <  U  <->  ( F `  r )  <  U
) )
3635, 21elrab2 2838 . . . . . . . . 9  |-  ( r  e.  L  <->  ( r  e.  ( A [,] B
)  /\  ( F `  r )  <  U
) )
3736simplbi 272 . . . . . . . 8  |-  ( r  e.  L  ->  r  e.  ( A [,] B
) )
3837ad2antlr 480 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  r  e.  ( A [,] B ) )
3933, 30, 38rspcdva 2789 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  e.  RR )
405ad3antrrr 483 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  U  e.  RR )
41 simpr 109 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  <  r
)
42 breq2 3928 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  <  y  <->  q  <  r ) )
43 fveq2 5414 . . . . . . . . . 10  |-  ( y  =  r  ->  ( F `  y )  =  ( F `  r ) )
4443breq2d 3936 . . . . . . . . 9  |-  ( y  =  r  ->  (
( F `  q
)  <  ( F `  y )  <->  ( F `  q )  <  ( F `  r )
) )
4542, 44imbi12d 233 . . . . . . . 8  |-  ( y  =  r  ->  (
( q  <  y  ->  ( F `  q
)  <  ( F `  y ) )  <->  ( q  <  r  ->  ( F `  q )  <  ( F `  r )
) ) )
46 breq1 3927 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <  y  <->  q  <  y ) )
4727breq1d 3934 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
( F `  x
)  <  ( F `  y )  <->  ( F `  q )  <  ( F `  y )
) )
4846, 47imbi12d 233 . . . . . . . . . 10  |-  ( x  =  q  ->  (
( x  <  y  ->  ( F `  x
)  <  ( F `  y ) )  <->  ( q  <  y  ->  ( F `  q )  <  ( F `  y )
) ) )
4948ralbidv 2435 . . . . . . . . 9  |-  ( x  =  q  ->  ( A. y  e.  ( A [,] B ) ( x  <  y  -> 
( F `  x
)  <  ( F `  y ) )  <->  A. y  e.  ( A [,] B
) ( q  < 
y  ->  ( F `  q )  <  ( F `  y )
) ) )
5018expr 372 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  y  e.  ( A [,] B
) )  ->  (
x  <  y  ->  ( F `  x )  <  ( F `  y ) ) )
5150ralrimiva 2503 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A. y  e.  ( A [,] B
) ( x  < 
y  ->  ( F `  x )  <  ( F `  y )
) )
5251ralrimiva 2503 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
)  <  ( F `  y ) ) )
5352ad3antrrr 483 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
)  <  ( F `  y ) ) )
5449, 53, 26rspcdva 2789 . . . . . . . 8  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  A. y  e.  ( A [,] B ) ( q  <  y  ->  ( F `  q
)  <  ( F `  y ) ) )
5545, 54, 38rspcdva 2789 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( q  < 
r  ->  ( F `  q )  <  ( F `  r )
) )
5641, 55mpd 13 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  <  ( F `  r )
)
5736simprbi 273 . . . . . . 7  |-  ( r  e.  L  ->  ( F `  r )  <  U )
5857ad2antlr 480 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  <  U
)
5931, 39, 40, 56, 58lttrd 7881 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  <  U
)
60 fveq2 5414 . . . . . . 7  |-  ( w  =  q  ->  ( F `  w )  =  ( F `  q ) )
6160breq1d 3934 . . . . . 6  |-  ( w  =  q  ->  (
( F `  w
)  <  U  <->  ( F `  q )  <  U
) )
6261, 21elrab2 2838 . . . . 5  |-  ( q  e.  L  <->  ( q  e.  ( A [,] B
)  /\  ( F `  q )  <  U
) )
6326, 59, 62sylanbrc 413 . . . 4  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  L
)
6463rexlimdva2 2550 . . 3  |-  ( (
ph  /\  q  e.  ( A [,] B ) )  ->  ( E. r  e.  L  q  <  r  ->  q  e.  L ) )
6525, 64impbid 128 . 2  |-  ( (
ph  /\  q  e.  ( A [,] B ) )  ->  ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
6665ralrimiva 2503 1  |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418    C_ wss 3066   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612    < clt 7793   [,]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-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  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-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-map 6537  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-rp 9435  df-icc 9671  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-cncf 12716
This theorem is referenced by:  ivthinclemex  12778
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