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Theorem ltxr 10704
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
ltxr  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( (
( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  =  -oo  /\  B  =  +oo ) )  \/  ( ( A  e.  RR  /\  B  = 
+oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )

Proof of Theorem ltxr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4209 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 938 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4264 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  =  { <. x ,  y
>.  |  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }
41, 3brab2ga 4942 . . . 4  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
54a1i 11 . . 3  |-  ( ( 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 ) ) )
6 brun 4250 . . . 4  |-  ( A ( ( ( RR  u.  {  -oo }
)  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  <->  ( A
( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  \/  A ( {  -oo }  X.  RR ) B ) )
7 brxp 4900 . . . . . . 7  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  <->  ( A  e.  ( RR  u.  {  -oo } )  /\  B  e.  {  +oo } ) )
8 elun 3480 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  RR  \/  A  e. 
{  -oo } ) )
9 orcom 377 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  {  -oo }
)  <->  ( A  e. 
{  -oo }  \/  A  e.  RR ) )
108, 9bitri 241 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  {  -oo }  \/  A  e.  RR )
)
11 elsncg 3828 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  {  -oo }  <->  A  =  -oo ) )
1211orbi1d 684 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  \/  A  e.  RR ) 
<->  ( A  =  -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 249 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  =  -oo  \/  A  e.  RR ) ) )
14 elsncg 3828 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  {  +oo }  <->  B  =  +oo ) )
1513, 14bi2anan9 844 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  {  -oo } )  /\  B  e. 
{  +oo } )  <->  ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) ) )
16 andir 839 . . . . . . . 8  |-  ( ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) 
<->  ( ( A  = 
-oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) ) )
1715, 16syl6bb 253 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  {  -oo } )  /\  B  e. 
{  +oo } )  <->  ( ( A  =  -oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) ) ) )
187, 17syl5bb 249 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( ( RR  u.  {  -oo }
)  X.  {  +oo } ) B  <->  ( ( A  =  -oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) ) ) )
19 brxp 4900 . . . . . . 7  |-  ( A ( {  -oo }  X.  RR ) B  <->  ( A  e.  {  -oo }  /\  B  e.  RR )
)
2011anbi1d 686 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  /\  B  e.  RR ) 
<->  ( A  =  -oo  /\  B  e.  RR ) ) )
2120adantr 452 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  {  -oo }  /\  B  e.  RR )  <->  ( A  =  -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 249 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( {  -oo }  X.  RR ) B  <-> 
( A  =  -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 691 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A ( ( RR  u.  {  -oo } )  X.  {  +oo } ) B  \/  A
( {  -oo }  X.  RR ) B )  <-> 
( ( ( A  =  -oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) )
24 orass 511 . . . . 5  |-  ( ( ( ( A  = 
-oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) )  \/  ( A  =  -oo  /\  B  e.  RR ) )  <->  ( ( A  =  -oo  /\  B  =  +oo )  \/  (
( A  e.  RR  /\  B  =  +oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) )
2523, 24syl6bb 253 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A ( ( RR  u.  {  -oo } )  X.  {  +oo } ) B  \/  A
( {  -oo }  X.  RR ) B )  <-> 
( ( A  = 
-oo  /\  B  =  +oo )  \/  (
( A  e.  RR  /\  B  =  +oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
266, 25syl5bb 249 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( { 
-oo }  X.  RR ) ) B  <->  ( ( A  =  -oo  /\  B  =  +oo )  \/  (
( A  e.  RR  /\  B  =  +oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
275, 26orbi12d 691 . 2  |-  ( ( 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  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  (
( A  =  -oo  /\  B  =  +oo )  \/  ( ( A  e.  RR  /\  B  = 
+oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) ) )
28 df-ltxr 9114 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
2928breqi 4210 . . 3  |-  ( 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 )
30 brun 4250 . . 3  |-  ( 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 ) )
3129, 30bitri 241 . 2  |-  ( 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 ) )
32 orass 511 . 2  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  =  -oo  /\  B  =  +oo ) )  \/  ( ( A  e.  RR  /\  B  = 
+oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  (
( A  =  -oo  /\  B  =  +oo )  \/  ( ( A  e.  RR  /\  B  = 
+oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
3327, 31, 323bitr4g 280 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( (
( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  =  -oo  /\  B  =  +oo ) )  \/  ( ( A  e.  RR  /\  B  = 
+oo )  \/  ( A  =  -oo  /\  B  e.  RR ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3310   {csn 3806   class class class wbr 4204   {copab 4257    X. cxp 4867   RRcr 8978    <RR cltrr 8983    +oocpnf 9106    -oocmnf 9107   RR*cxr 9108    < clt 9109
This theorem is referenced by:  xrltnr  10709  ltpnf  10710  mnflt  10711  mnfltpnf  10712  pnfnlt  10714  nltmnf  10715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-ltxr 9114
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