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Theorem ivthinclemloc 13413
Description: Lemma for ivthinc 13415. Locatedness. (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
ivthinclemloc  |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  R
) ) )
Distinct variable groups:    A, r, w   
x, A, y, r    B, r, w    x, B, y    w, F    x, F, 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( x, y, w, r, q)

Proof of Theorem ivthinclemloc
StepHypRef Expression
1 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  q  <  r )
2 breq2 3993 . . . . . . . 8  |-  ( y  =  r  ->  (
q  <  y  <->  q  <  r ) )
3 fveq2 5496 . . . . . . . . 9  |-  ( y  =  r  ->  ( F `  y )  =  ( F `  r ) )
43breq2d 4001 . . . . . . . 8  |-  ( y  =  r  ->  (
( F `  q
)  <  ( F `  y )  <->  ( F `  q )  <  ( F `  r )
) )
52, 4imbi12d 233 . . . . . . 7  |-  ( y  =  r  ->  (
( q  <  y  ->  ( F `  q
)  <  ( F `  y ) )  <->  ( q  <  r  ->  ( F `  q )  <  ( F `  r )
) ) )
6 breq1 3992 . . . . . . . . . 10  |-  ( x  =  q  ->  (
x  <  y  <->  q  <  y ) )
7 fveq2 5496 . . . . . . . . . . 11  |-  ( x  =  q  ->  ( F `  x )  =  ( F `  q ) )
87breq1d 3999 . . . . . . . . . 10  |-  ( x  =  q  ->  (
( F `  x
)  <  ( F `  y )  <->  ( F `  q )  <  ( F `  y )
) )
96, 8imbi12d 233 . . . . . . . . 9  |-  ( x  =  q  ->  (
( x  <  y  ->  ( F `  x
)  <  ( F `  y ) )  <->  ( q  <  y  ->  ( F `  q )  <  ( F `  y )
) ) )
109ralbidv 2470 . . . . . . . 8  |-  ( x  =  q  ->  ( A. y  e.  ( A [,] B ) ( x  <  y  -> 
( F `  x
)  <  ( F `  y ) )  <->  A. y  e.  ( A [,] B
) ( q  < 
y  ->  ( F `  q )  <  ( F `  y )
) ) )
11 ivthinc.i . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  (
y  e.  ( A [,] B )  /\  x  <  y ) )  ->  ( F `  x )  <  ( F `  y )
)
1211expr 373 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  ( A [,] B
) )  /\  y  e.  ( A [,] B
) )  ->  (
x  <  y  ->  ( F `  x )  <  ( F `  y ) ) )
1312ralrimiva 2543 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  A. y  e.  ( A [,] B
) ( x  < 
y  ->  ( F `  x )  <  ( F `  y )
) )
1413ralrimiva 2543 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
)  <  ( F `  y ) ) )
1514ad2antrr 485 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  A. x  e.  ( A [,] B
) A. y  e.  ( A [,] B
) ( x  < 
y  ->  ( F `  x )  <  ( F `  y )
) )
16 simplrl 530 . . . . . . . 8  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  q  e.  ( A [,] B ) )
1710, 15, 16rspcdva 2839 . . . . . . 7  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  A. y  e.  ( A [,] B
) ( q  < 
y  ->  ( F `  q )  <  ( F `  y )
) )
18 simplrr 531 . . . . . . 7  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  r  e.  ( A [,] B ) )
195, 17, 18rspcdva 2839 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( q  <  r  ->  ( F `  q )  <  ( F `  r )
) )
201, 19mpd 13 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( F `  q )  <  ( F `  r )
)
217eleq1d 2239 . . . . . . 7  |-  ( x  =  q  ->  (
( F `  x
)  e.  RR  <->  ( F `  q )  e.  RR ) )
22 ivth.8 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( A [,] B ) )  ->  ( F `  x )  e.  RR )
2322ralrimiva 2543 . . . . . . . 8  |-  ( ph  ->  A. x  e.  ( A [,] B ) ( F `  x
)  e.  RR )
2423ad2antrr 485 . . . . . . 7  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  A. x  e.  ( A [,] B
) ( F `  x )  e.  RR )
2521, 24, 16rspcdva 2839 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( F `  q )  e.  RR )
26 fveq2 5496 . . . . . . . 8  |-  ( x  =  r  ->  ( F `  x )  =  ( F `  r ) )
2726eleq1d 2239 . . . . . . 7  |-  ( x  =  r  ->  (
( F `  x
)  e.  RR  <->  ( F `  r )  e.  RR ) )
2827, 24, 18rspcdva 2839 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( F `  r )  e.  RR )
29 ivth.3 . . . . . . 7  |-  ( ph  ->  U  e.  RR )
3029ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  U  e.  RR )
31 axltwlin 7987 . . . . . 6  |-  ( ( ( F `  q
)  e.  RR  /\  ( F `  r )  e.  RR  /\  U  e.  RR )  ->  (
( F `  q
)  <  ( F `  r )  ->  (
( F `  q
)  <  U  \/  U  <  ( F `  r ) ) ) )
3225, 28, 30, 31syl3anc 1233 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( ( F `  q )  <  ( F `  r
)  ->  ( ( F `  q )  <  U  \/  U  < 
( F `  r
) ) ) )
3320, 32mpd 13 . . . 4  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( ( F `  q )  <  U  \/  U  < 
( F `  r
) ) )
3416adantr 274 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  ( F `  q )  <  U )  ->  q  e.  ( A [,] B
) )
35 simpr 109 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  ( F `  q )  <  U )  ->  ( F `  q )  <  U )
36 fveq2 5496 . . . . . . . . 9  |-  ( w  =  q  ->  ( F `  w )  =  ( F `  q ) )
3736breq1d 3999 . . . . . . . 8  |-  ( w  =  q  ->  (
( F `  w
)  <  U  <->  ( F `  q )  <  U
) )
38 ivthinclem.l . . . . . . . 8  |-  L  =  { w  e.  ( A [,] B )  |  ( F `  w )  <  U }
3937, 38elrab2 2889 . . . . . . 7  |-  ( q  e.  L  <->  ( q  e.  ( A [,] B
)  /\  ( F `  q )  <  U
) )
4034, 35, 39sylanbrc 415 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  ( F `  q )  <  U )  ->  q  e.  L )
4140ex 114 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( ( F `  q )  <  U  ->  q  e.  L ) )
4218adantr 274 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  U  <  ( F `  r
) )  ->  r  e.  ( A [,] B
) )
43 simpr 109 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  U  <  ( F `  r
) )  ->  U  <  ( F `  r
) )
44 fveq2 5496 . . . . . . . . 9  |-  ( w  =  r  ->  ( F `  w )  =  ( F `  r ) )
4544breq2d 4001 . . . . . . . 8  |-  ( w  =  r  ->  ( U  <  ( F `  w )  <->  U  <  ( F `  r ) ) )
46 ivthinclem.r . . . . . . . 8  |-  R  =  { w  e.  ( A [,] B )  |  U  <  ( F `  w ) }
4745, 46elrab2 2889 . . . . . . 7  |-  ( r  e.  R  <->  ( r  e.  ( A [,] B
)  /\  U  <  ( F `  r ) ) )
4842, 43, 47sylanbrc 415 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r )  /\  U  <  ( F `  r
) )  ->  r  e.  R )
4948ex 114 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( U  <  ( F `  r
)  ->  r  e.  R ) )
5041, 49orim12d 781 . . . 4  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( (
( F `  q
)  <  U  \/  U  <  ( F `  r ) )  -> 
( q  e.  L  \/  r  e.  R
) ) )
5133, 50mpd 13 . . 3  |-  ( ( ( ph  /\  (
q  e.  ( A [,] B )  /\  r  e.  ( A [,] B ) ) )  /\  q  <  r
)  ->  ( q  e.  L  \/  r  e.  R ) )
5251ex 114 . 2  |-  ( (
ph  /\  ( q  e.  ( A [,] B
)  /\  r  e.  ( A [,] B ) ) )  ->  (
q  <  r  ->  ( q  e.  L  \/  r  e.  R )
) )
5352ralrimivva 2552 1  |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  R
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 703    = wceq 1348    e. wcel 2141   A.wral 2448   {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-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltwlin 7887
This theorem depends on definitions:  df-bi 116  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-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-iota 5160  df-fv 5206  df-pnf 7956  df-mnf 7957  df-ltxr 7959
This theorem is referenced by:  ivthinclemex  13414
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