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Theorem ivthinclemlr 13015
Description: Lemma for ivthinc 13021. 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 480 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  e.  RR )
3 ivth.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
43ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  B  e.  RR )
5 ivth.3 . . . . . 6  |-  ( ph  ->  U  e.  RR )
65ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  U  e.  RR )
7 ivth.4 . . . . . 6  |-  ( ph  ->  A  <  B )
87ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  <  B )
9 ivth.5 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  D )
109ad2antrr 480 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  ( A [,] B )  C_  D )
11 ivth.7 . . . . . 6  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1211ad2antrr 480 . . . . 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 469 . . . . . 6  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  x  e.  ( A [,] B
) )  ->  ( F `  x )  e.  RR )
1514adantlr 469 . . . . 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 480 . . . . 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 473 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  <  y
) )  ->  ( F `  x )  <  ( F `  y
) )
2019adantllr 473 . . . . 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 13014 . . . 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 524 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  ( A [,] B ) )
27 fveq2 5468 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
2827eleq1d 2226 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  e.  RR  <->  ( F `  q )  e.  RR ) )
2913ralrimiva 2530 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3029ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3128, 30, 26rspcdva 2821 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  e.  RR )
32 fveq2 5468 . . . . . . . 8  |-  ( x  =  r  ->  ( F `  x )  =  ( F `  r ) )
3332eleq1d 2226 . . . . . . 7  |-  ( x  =  r  ->  (
( F `  x
)  e.  RR  <->  ( F `  r )  e.  RR ) )
34 fveq2 5468 . . . . . . . . . . 11  |-  ( w  =  r  ->  ( F `  w )  =  ( F `  r ) )
3534breq1d 3975 . . . . . . . . . 10  |-  ( w  =  r  ->  (
( F `  w
)  <  U  <->  ( F `  r )  <  U
) )
3635, 21elrab2 2871 . . . . . . . . 9  |-  ( r  e.  L  <->  ( r  e.  ( A [,] B
)  /\  ( F `  r )  <  U
) )
3736simplbi 272 . . . . . . . 8  |-  ( r  e.  L  ->  r  e.  ( A [,] B
) )
3837ad2antlr 481 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  r  e.  ( A [,] B ) )
3933, 30, 38rspcdva 2821 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  e.  RR )
405ad3antrrr 484 . . . . . 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 3969 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  <  y  <->  q  <  r ) )
43 fveq2 5468 . . . . . . . . . 10  |-  ( y  =  r  ->  ( F `  y )  =  ( F `  r ) )
4443breq2d 3977 . . . . . . . . 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 3968 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <  y  <->  q  <  y ) )
4727breq1d 3975 . . . . . . . . . . 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 2457 . . . . . . . . 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 373 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  y  e.  ( A [,] B
) )  ->  (
x  <  y  ->  ( F `  x )  <  ( F `  y ) ) )
5150ralrimiva 2530 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A. y  e.  ( A [,] B
) ( x  < 
y  ->  ( F `  x )  <  ( F `  y )
) )
5251ralrimiva 2530 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
)  <  ( F `  y ) ) )
5352ad3antrrr 484 . . . . . . . . 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 2821 . . . . . . . 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 2821 . . . . . . 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 481 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  <  U
)
5931, 39, 40, 56, 58lttrd 8001 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  <  U
)
60 fveq2 5468 . . . . . . 7  |-  ( w  =  q  ->  ( F `  w )  =  ( F `  q ) )
6160breq1d 3975 . . . . . 6  |-  ( w  =  q  ->  (
( F `  w
)  <  U  <->  ( F `  q )  <  U
) )
6261, 21elrab2 2871 . . . . 5  |-  ( q  e.  L  <->  ( q  e.  ( A [,] B
)  /\  ( F `  q )  <  U
) )
6326, 59, 62sylanbrc 414 . . . 4  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  L
)
6463rexlimdva2 2577 . . 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 2530 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 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439    C_ wss 3102   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   CCcc 7730   RRcr 7731    < clt 7912   [,]cicc 9795   -cn->ccncf 12957
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-map 6595  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-icc 9799  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-cncf 12958
This theorem is referenced by:  ivthinclemex  13020
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