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Theorem ivthinclemlr 13409
Description: Lemma for ivthinc 13415. 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 485 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  e.  RR )
3 ivth.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
43ad2antrr 485 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  B  e.  RR )
5 ivth.3 . . . . . 6  |-  ( ph  ->  U  e.  RR )
65ad2antrr 485 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  U  e.  RR )
7 ivth.4 . . . . . 6  |-  ( ph  ->  A  <  B )
87ad2antrr 485 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  A  <  B )
9 ivth.5 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  D )
109ad2antrr 485 . . . . 5  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  q  e.  L )  ->  ( A [,] B )  C_  D )
11 ivth.7 . . . . . 6  |-  ( ph  ->  F  e.  ( D
-cn-> CC ) )
1211ad2antrr 485 . . . . 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 474 . . . . . 6  |-  ( ( ( ph  /\  q  e.  ( A [,] B
) )  /\  x  e.  ( A [,] B
) )  ->  ( F `  x )  e.  RR )
1514adantlr 474 . . . . 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 485 . . . . 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 478 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  x  e.  ( A [,] B ) )  /\  ( y  e.  ( A [,] B )  /\  x  <  y
) )  ->  ( F `  x )  <  ( F `  y
) )
2019adantllr 478 . . . . 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 13408 . . . 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 529 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  ( A [,] B ) )
27 fveq2 5496 . . . . . . . 8  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
2827eleq1d 2239 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  e.  RR  <->  ( F `  q )  e.  RR ) )
2913ralrimiva 2543 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3029ad3antrrr 489 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
3128, 30, 26rspcdva 2839 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  e.  RR )
32 fveq2 5496 . . . . . . . 8  |-  ( x  =  r  ->  ( F `  x )  =  ( F `  r ) )
3332eleq1d 2239 . . . . . . 7  |-  ( x  =  r  ->  (
( F `  x
)  e.  RR  <->  ( F `  r )  e.  RR ) )
34 fveq2 5496 . . . . . . . . . . 11  |-  ( w  =  r  ->  ( F `  w )  =  ( F `  r ) )
3534breq1d 3999 . . . . . . . . . 10  |-  ( w  =  r  ->  (
( F `  w
)  <  U  <->  ( F `  r )  <  U
) )
3635, 21elrab2 2889 . . . . . . . . 9  |-  ( r  e.  L  <->  ( r  e.  ( A [,] B
)  /\  ( F `  r )  <  U
) )
3736simplbi 272 . . . . . . . 8  |-  ( r  e.  L  ->  r  e.  ( A [,] B
) )
3837ad2antlr 486 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  r  e.  ( A [,] B ) )
3933, 30, 38rspcdva 2839 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  e.  RR )
405ad3antrrr 489 . . . . . 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 3993 . . . . . . . . 9  |-  ( y  =  r  ->  (
q  <  y  <->  q  <  r ) )
43 fveq2 5496 . . . . . . . . . 10  |-  ( y  =  r  ->  ( F `  y )  =  ( F `  r ) )
4443breq2d 4001 . . . . . . . . 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 3992 . . . . . . . . . . 11  |-  ( x  =  q  ->  (
x  <  y  <->  q  <  y ) )
4727breq1d 3999 . . . . . . . . . . 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 2470 . . . . . . . . 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 2543 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A. y  e.  ( A [,] B
) ( x  < 
y  ->  ( F `  x )  <  ( F `  y )
) )
5251ralrimiva 2543 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
)  <  ( F `  y ) ) )
5352ad3antrrr 489 . . . . . . . . 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 2839 . . . . . . . 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 2839 . . . . . . 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 486 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  r )  <  U
)
5931, 39, 40, 56, 58lttrd 8045 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  ( F `  q )  <  U
)
60 fveq2 5496 . . . . . . 7  |-  ( w  =  q  ->  ( F `  w )  =  ( F `  q ) )
6160breq1d 3999 . . . . . 6  |-  ( w  =  q  ->  (
( F `  w
)  <  U  <->  ( F `  q )  <  U
) )
6261, 21elrab2 2889 . . . . 5  |-  ( q  e.  L  <->  ( q  e.  ( A [,] B
)  /\  ( F `  q )  <  U
) )
6326, 59, 62sylanbrc 415 . . . 4  |-  ( ( ( ( ph  /\  q  e.  ( A [,] B ) )  /\  r  e.  L )  /\  q  <  r )  ->  q  e.  L
)
6463rexlimdva2 2590 . . 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 2543 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 1348    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452    C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773    < clt 7954   [,]cicc 9848   -cn->ccncf 13351
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-map 6628  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-rp 9611  df-icc 9852  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-cncf 13352
This theorem is referenced by:  ivthinclemex  13414
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