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Theorem ltxrlt 7830
Description: The standard less-than  <RR and the extended real less-than  < are identical when restricted to the non-extended reals  RR. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ltxrlt  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )

Proof of Theorem ltxrlt
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltxr 7805 . . . . 5  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
21breqi 3935 . . . 4  |-  ( A  <  B  <->  A ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) ) B )
3 brun 3979 . . . 4  |-  ( A ( { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  ( ( ( RR  u.  { -oo }
)  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) ) B  <-> 
( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B ) )
42, 3bitri 183 . . 3  |-  ( A  <  B  <->  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B ) )
5 eleq1 2202 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  RR  <->  A  e.  RR ) )
6 breq1 3932 . . . . . . 7  |-  ( x  =  A  ->  (
x  <RR  y  <->  A  <RR  y ) )
75, 63anbi13d 1292 . . . . . 6  |-  ( x  =  A  ->  (
( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( A  e.  RR  /\  y  e.  RR  /\  A  <RR  y ) ) )
8 eleq1 2202 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  RR  <->  B  e.  RR ) )
9 breq2 3933 . . . . . . 7  |-  ( y  =  B  ->  ( A  <RR  y  <->  A  <RR  B ) )
108, 93anbi23d 1293 . . . . . 6  |-  ( y  =  B  ->  (
( A  e.  RR  /\  y  e.  RR  /\  A  <RR  y )  <->  ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B ) ) )
11 eqid 2139 . . . . . 6  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  =  { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }
127, 10, 11brabg 4191 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B ) ) )
13 simp3 983 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A  <RR  B )
1412, 13syl6bi 162 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  ->  A  <RR  B ) )
15 brun 3979 . . . . 5  |-  ( A ( ( ( RR  u.  { -oo }
)  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B  <->  ( A
( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  \/  A ( { -oo }  X.  RR ) B ) )
16 brxp 4570 . . . . . . . . . . 11  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
1716simprbi 273 . . . . . . . . . 10  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  ->  B  e.  { +oo } )
18 elsni 3545 . . . . . . . . . 10  |-  ( B  e.  { +oo }  ->  B  = +oo )
1917, 18syl 14 . . . . . . . . 9  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  ->  B  = +oo )
2019a1i 9 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A ( ( RR  u.  { -oo }
)  X.  { +oo } ) B  ->  B  = +oo ) )
21 renepnf 7813 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  =/= +oo )
2221neneqd 2329 . . . . . . . 8  |-  ( B  e.  RR  ->  -.  B  = +oo )
23 pm2.24 610 . . . . . . . 8  |-  ( B  = +oo  ->  ( -.  B  = +oo  ->  A  <RR  B ) )
2420, 22, 23syl6ci 1421 . . . . . . 7  |-  ( B  e.  RR  ->  ( A ( ( RR  u.  { -oo }
)  X.  { +oo } ) B  ->  A  <RR  B ) )
2524adantl 275 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  ->  A  <RR  B ) )
26 brxp 4570 . . . . . . . . . . 11  |-  ( A ( { -oo }  X.  RR ) B  <->  ( A  e.  { -oo }  /\  B  e.  RR )
)
2726simplbi 272 . . . . . . . . . 10  |-  ( A ( { -oo }  X.  RR ) B  ->  A  e.  { -oo }
)
28 elsni 3545 . . . . . . . . . 10  |-  ( A  e.  { -oo }  ->  A  = -oo )
2927, 28syl 14 . . . . . . . . 9  |-  ( A ( { -oo }  X.  RR ) B  ->  A  = -oo )
3029a1i 9 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A ( { -oo }  X.  RR ) B  ->  A  = -oo ) )
31 renemnf 7814 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
3231neneqd 2329 . . . . . . . 8  |-  ( A  e.  RR  ->  -.  A  = -oo )
33 pm2.24 610 . . . . . . . 8  |-  ( A  = -oo  ->  ( -.  A  = -oo  ->  A  <RR  B ) )
3430, 32, 33syl6ci 1421 . . . . . . 7  |-  ( A  e.  RR  ->  ( A ( { -oo }  X.  RR ) B  ->  A  <RR  B ) )
3534adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( { -oo }  X.  RR ) B  ->  A  <RR  B ) )
3625, 35jaod 706 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  \/  A ( { -oo }  X.  RR ) B )  ->  A  <RR  B ) )
3715, 36syl5bi 151 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B  ->  A  <RR  B ) )
3814, 37jaod 706 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B )  ->  A  <RR  B ) )
394, 38syl5bi 151 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <RR  B ) )
40123adant3 1001 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B ) ) )
4140ibir 176 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B )
4241orcd 722 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B ) )
4342, 4sylibr 133 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A  <  B )
44433expia 1183 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  ->  A  <  B ) )
4539, 44impbid 128 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480    u. cun 3069   {csn 3527   class class class wbr 3929   {copab 3988    X. cxp 4537   RRcr 7619    <RR cltrr 7624   +oocpnf 7797   -oocmnf 7798    < clt 7800
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-ltxr 7805
This theorem is referenced by:  axltirr  7831  axltwlin  7832  axlttrn  7833  axltadd  7834  axapti  7835  axmulgt0  7836  axsuploc  7837  0lt1  7889  recexre  8340  recexgt0  8342  remulext1  8361  arch  8974  caucvgrelemcau  10752  caucvgre  10753
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