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Theorem ltxrlt 8893
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 brun 4069 . . . . 5  |-  ( A ( ( ( RR  u.  {  -oo }
)  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  <->  ( A
( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  \/  A ( {  -oo }  X.  RR ) B ) )
2 brxp 4720 . . . . . . 7  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  <->  ( A  e.  ( RR  u.  {  -oo } )  /\  B  e.  {  +oo } ) )
3 elsni 3664 . . . . . . . . 9  |-  ( B  e.  {  +oo }  ->  B  =  +oo )
4 pnfnre 8874 . . . . . . . . . . 11  |-  +oo  e/  RR
5 df-nel 2449 . . . . . . . . . . 11  |-  (  +oo  e/  RR  <->  -.  +oo  e.  RR )
64, 5mpbi 199 . . . . . . . . . 10  |-  -.  +oo  e.  RR
7 simpr 447 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
8 eleq1 2343 . . . . . . . . . . 11  |-  ( B  =  +oo  ->  ( B  e.  RR  <->  +oo  e.  RR ) )
97, 8syl5ib 210 . . . . . . . . . 10  |-  ( B  =  +oo  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  +oo  e.  RR ) )
106, 9mtoi 169 . . . . . . . . 9  |-  ( B  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
113, 10syl 15 . . . . . . . 8  |-  ( B  e.  {  +oo }  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1211adantl 452 . . . . . . 7  |-  ( ( A  e.  ( RR  u.  {  -oo }
)  /\  B  e.  { 
+oo } )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
132, 12sylbi 187 . . . . . 6  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 brxp 4720 . . . . . . 7  |-  ( A ( {  -oo }  X.  RR ) B  <->  ( A  e.  {  -oo }  /\  B  e.  RR )
)
15 elsni 3664 . . . . . . . . 9  |-  ( A  e.  {  -oo }  ->  A  =  -oo )
16 mnfnre 8875 . . . . . . . . . . 11  |-  -oo  e/  RR
17 df-nel 2449 . . . . . . . . . . 11  |-  (  -oo  e/  RR  <->  -.  -oo  e.  RR )
1816, 17mpbi 199 . . . . . . . . . 10  |-  -.  -oo  e.  RR
19 simpl 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
20 eleq1 2343 . . . . . . . . . . 11  |-  ( A  =  -oo  ->  ( A  e.  RR  <->  -oo  e.  RR ) )
2119, 20syl5ib 210 . . . . . . . . . 10  |-  ( A  =  -oo  ->  (
( A  e.  RR  /\  B  e.  RR )  ->  -oo  e.  RR ) )
2218, 21mtoi 169 . . . . . . . . 9  |-  ( A  =  -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2315, 22syl 15 . . . . . . . 8  |-  ( A  e.  {  -oo }  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2423adantr 451 . . . . . . 7  |-  ( ( A  e.  {  -oo }  /\  B  e.  RR )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2514, 24sylbi 187 . . . . . 6  |-  ( A ( {  -oo }  X.  RR ) B  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2613, 25jaoi 368 . . . . 5  |-  ( ( A ( ( RR  u.  {  -oo }
)  X.  {  +oo } ) B  \/  A
( {  -oo }  X.  RR ) B )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
271, 26sylbi 187 . . . 4  |-  ( A ( ( ( RR  u.  {  -oo }
)  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2827con2i 112 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  A ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) B )
29 biimt 325 . . . 4  |-  ( -.  A ( ( ( 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  ->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) ) )
30 df-ltxr 8872 . . . . . . 7  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
3130equncomi 3321 . . . . . 6  |-  <  =  ( ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) )  u.  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } )
3231breqi 4029 . . . . 5  |-  ( A  <  B  <->  A (
( ( ( RR  u.  {  -oo }
)  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) )  u.  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } ) B )
33 brun 4069 . . . . 5  |-  ( A ( ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) )  u.  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } ) B  <-> 
( A ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) B  \/  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
34 df-or 359 . . . . 5  |-  ( ( A ( ( ( 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  ->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
3532, 33, 343bitri 262 . . . 4  |-  ( A  <  B  <->  ( -.  A ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  ->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
3629, 35syl6rbbr 255 . . 3  |-  ( -.  A ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  -> 
( A  <  B  <->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
3728, 36syl 15 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
38 breq12 4028 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
39 df-3an 936 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
4039opabbii 4083 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  =  { <. x ,  y
>.  |  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }
4138, 40brab2ga 4763 . . 3  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
4241baibr 872 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B ) )
4337, 42bitr4d 247 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    e/ wnel 2447    u. cun 3150   {csn 3640   class class class wbr 4023   {copab 4076    X. cxp 4687   RRcr 8736    <RR cltrr 8741    +oocpnf 8864    -oocmnf 8865    < clt 8867
This theorem is referenced by:  axlttri  8894  axlttrn  8895  axltadd  8896  axmulgt0  8897  axsup  8898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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