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Theorem ltxrlt 7245
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 7220 . . . . 5  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
21breqi 3799 . . . 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 3839 . . . 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 182 . . 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 2142 . . . . . . 7  |-  ( x  =  A  ->  (
x  e.  RR  <->  A  e.  RR ) )
6 breq1 3796 . . . . . . 7  |-  ( x  =  A  ->  (
x  <RR  y  <->  A  <RR  y ) )
75, 63anbi13d 1246 . . . . . 6  |-  ( x  =  A  ->  (
( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( A  e.  RR  /\  y  e.  RR  /\  A  <RR  y ) ) )
8 eleq1 2142 . . . . . . 7  |-  ( y  =  B  ->  (
y  e.  RR  <->  B  e.  RR ) )
9 breq2 3797 . . . . . . 7  |-  ( y  =  B  ->  ( A  <RR  y  <->  A  <RR  B ) )
108, 93anbi23d 1247 . . . . . 6  |-  ( y  =  B  ->  (
( A  e.  RR  /\  y  e.  RR  /\  A  <RR  y )  <->  ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B ) ) )
11 eqid 2082 . . . . . 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 4032 . . . . 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 941 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A  <RR  B )
1412, 13syl6bi 161 . . . 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 3839 . . . . 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 4401 . . . . . . . . . . 11  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
1716simprbi 269 . . . . . . . . . 10  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  ->  B  e.  { +oo } )
18 elsni 3424 . . . . . . . . . 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 7228 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  =/= +oo )
2221neneqd 2267 . . . . . . . 8  |-  ( B  e.  RR  ->  -.  B  = +oo )
23 pm2.24 584 . . . . . . . 8  |-  ( B  = +oo  ->  ( -.  B  = +oo  ->  A  <RR  B ) )
2420, 22, 23syl6ci 1375 . . . . . . 7  |-  ( B  e.  RR  ->  ( A ( ( RR  u.  { -oo }
)  X.  { +oo } ) B  ->  A  <RR  B ) )
2524adantl 271 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  ->  A  <RR  B ) )
26 brxp 4401 . . . . . . . . . . 11  |-  ( A ( { -oo }  X.  RR ) B  <->  ( A  e.  { -oo }  /\  B  e.  RR )
)
2726simplbi 268 . . . . . . . . . 10  |-  ( A ( { -oo }  X.  RR ) B  ->  A  e.  { -oo }
)
28 elsni 3424 . . . . . . . . . 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 7229 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =/= -oo )
3231neneqd 2267 . . . . . . . 8  |-  ( A  e.  RR  ->  -.  A  = -oo )
33 pm2.24 584 . . . . . . . 8  |-  ( A  = -oo  ->  ( -.  A  = -oo  ->  A  <RR  B ) )
3430, 32, 33syl6ci 1375 . . . . . . 7  |-  ( A  e.  RR  ->  ( A ( { -oo }  X.  RR ) B  ->  A  <RR  B ) )
3534adantr 270 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( { -oo }  X.  RR ) B  ->  A  <RR  B ) )
3625, 35jaod 670 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  \/  A ( { -oo }  X.  RR ) B )  ->  A  <RR  B ) )
3715, 36syl5bi 150 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B  ->  A  <RR  B ) )
3814, 37jaod 670 . . 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 150 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <RR  B ) )
40123adant3 959 . . . . . 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 175 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B )
4241orcd 685 . . . 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 132 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <RR  B )  ->  A  <  B )
44433expia 1141 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  ->  A  <  B ) )
4539, 44impbid 127 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 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434    u. cun 2972   {csn 3406   class class class wbr 3793   {copab 3846    X. cxp 4369   RRcr 7042    <RR cltrr 7047   +oocpnf 7212   -oocmnf 7213    < clt 7215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-pnf 7217  df-mnf 7218  df-ltxr 7220
This theorem is referenced by:  axltirr  7246  axltwlin  7247  axlttrn  7248  axltadd  7249  axapti  7250  axmulgt0  7251  0lt1  7303  recexre  7745  recexgt0  7747  remulext1  7766  arch  8352  caucvgrelemcau  10004  caucvgre  10005
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