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Theorem ltsosr 7596
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
Assertion
Ref Expression
ltsosr  |-  <R  Or  R.

Proof of Theorem ltsosr
Dummy variables  a  b  c  d  e  f  r  s  t  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltposr 7595 . 2  |-  <R  Po  R.
2 df-nr 7559 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
3 breq1 3940 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  x  <R  [
<. c ,  d >. ]  ~R  ) )
4 breq1 3940 . . . . . 6  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f
>. ]  ~R  <->  x  <R  [
<. e ,  f >. ]  ~R  ) )
54orbi1d 781 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( ( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f
>. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) 
<->  ( x  <R  [ <. e ,  f >. ]  ~R  \/  [ <. e ,  f
>. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) ) )
63, 5imbi12d 233 . . . 4  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( ( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  ->  ( [ <. a ,  b
>. ]  ~R  <R  [ <. e ,  f >. ]  ~R  \/  [ <. e ,  f
>. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) )  <->  ( x  <R  [ <. c ,  d
>. ]  ~R  ->  (
x  <R  [ <. e ,  f >. ]  ~R  \/  [ <. e ,  f
>. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) ) ) )
7 breq2 3941 . . . . 5  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( x  <R  [ <. c ,  d >. ]  ~R  <->  x 
<R  y ) )
8 breq2 3941 . . . . . 6  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  [ <. e ,  f >. ]  ~R  <R  y ) )
98orbi2d 780 . . . . 5  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( ( x  <R  [
<. e ,  f >. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  )  <->  ( x  <R  [ <. e ,  f
>. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y )
) )
107, 9imbi12d 233 . . . 4  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( ( x  <R  [
<. c ,  d >. ]  ~R  ->  ( x  <R  [ <. e ,  f
>. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) )  <->  ( x  <R  y  ->  ( x  <R  [ <. e ,  f
>. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y )
) ) )
11 breq2 3941 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( x  <R  [ <. e ,  f >. ]  ~R  <->  x 
<R  z ) )
12 breq1 3940 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( [ <. e ,  f >. ]  ~R  <R  y  <->  z  <R  y
) )
1311, 12orbi12d 783 . . . . 5  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( ( x  <R  [
<. e ,  f >. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y )  <->  ( x  <R  z  \/  z  <R 
y ) ) )
1413imbi2d 229 . . . 4  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( ( x  <R  y  ->  ( x  <R  [
<. e ,  f >. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y ) )  <->  ( x  <R  y  ->  ( x  <R  z  \/  z  <R 
y ) ) ) )
15 simp1l 1006 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  a  e.  P. )
16 simp3r 1011 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  f  e.  P. )
17 addclpr 7369 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  f  e.  P. )  ->  ( a  +P.  f
)  e.  P. )
1815, 16, 17syl2anc 409 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( a  +P.  f )  e.  P. )
19 simp2r 1009 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  d  e.  P. )
20 addclpr 7369 . . . . . . . 8  |-  ( ( ( a  +P.  f
)  e.  P.  /\  d  e.  P. )  ->  ( ( a  +P.  f )  +P.  d
)  e.  P. )
2118, 19, 20syl2anc 409 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  f )  +P.  d )  e.  P. )
22 simp2l 1008 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  c  e.  P. )
23 addclpr 7369 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  c  e.  P. )  ->  ( f  +P.  c
)  e.  P. )
2416, 22, 23syl2anc 409 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( f  +P.  c )  e.  P. )
25 simp1r 1007 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  b  e.  P. )
26 addclpr 7369 . . . . . . . 8  |-  ( ( ( f  +P.  c
)  e.  P.  /\  b  e.  P. )  ->  ( ( f  +P.  c )  +P.  b
)  e.  P. )
2724, 25, 26syl2anc 409 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
f  +P.  c )  +P.  b )  e.  P. )
28 simp3l 1010 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  e  e.  P. )
29 addclpr 7369 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  e  e.  P. )  ->  ( b  +P.  e
)  e.  P. )
3025, 28, 29syl2anc 409 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( b  +P.  e )  e.  P. )
31 addclpr 7369 . . . . . . . 8  |-  ( ( ( b  +P.  e
)  e.  P.  /\  d  e.  P. )  ->  ( ( b  +P.  e )  +P.  d
)  e.  P. )
3230, 19, 31syl2anc 409 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
b  +P.  e )  +P.  d )  e.  P. )
33 ltsopr 7428 . . . . . . . 8  |-  <P  Or  P.
34 sowlin 4250 . . . . . . . 8  |-  ( ( 
<P  Or  P.  /\  (
( ( a  +P.  f )  +P.  d
)  e.  P.  /\  ( ( f  +P.  c )  +P.  b
)  e.  P.  /\  ( ( b  +P.  e )  +P.  d
)  e.  P. )
)  ->  ( (
( a  +P.  f
)  +P.  d )  <P  ( ( f  +P.  c )  +P.  b
)  ->  ( (
( a  +P.  f
)  +P.  d )  <P  ( ( b  +P.  e )  +P.  d
)  \/  ( ( b  +P.  e )  +P.  d )  <P 
( ( f  +P.  c )  +P.  b
) ) ) )
3533, 34mpan 421 . . . . . . 7  |-  ( ( ( ( a  +P.  f )  +P.  d
)  e.  P.  /\  ( ( f  +P.  c )  +P.  b
)  e.  P.  /\  ( ( b  +P.  e )  +P.  d
)  e.  P. )  ->  ( ( ( a  +P.  f )  +P.  d )  <P  (
( f  +P.  c
)  +P.  b )  ->  ( ( ( a  +P.  f )  +P.  d )  <P  (
( b  +P.  e
)  +P.  d )  \/  ( ( b  +P.  e )  +P.  d
)  <P  ( ( f  +P.  c )  +P.  b ) ) ) )
3621, 27, 32, 35syl3anc 1217 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
( a  +P.  f
)  +P.  d )  <P  ( ( f  +P.  c )  +P.  b
)  ->  ( (
( a  +P.  f
)  +P.  d )  <P  ( ( b  +P.  e )  +P.  d
)  \/  ( ( b  +P.  e )  +P.  d )  <P 
( ( f  +P.  c )  +P.  b
) ) ) )
37 addclpr 7369 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  d  e.  P. )  ->  ( a  +P.  d
)  e.  P. )
3815, 19, 37syl2anc 409 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( a  +P.  d )  e.  P. )
39 addclpr 7369 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  c  e.  P. )  ->  ( b  +P.  c
)  e.  P. )
4025, 22, 39syl2anc 409 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( b  +P.  c )  e.  P. )
41 ltaprg 7451 . . . . . . . 8  |-  ( ( ( a  +P.  d
)  e.  P.  /\  ( b  +P.  c
)  e.  P.  /\  f  e.  P. )  ->  ( ( a  +P.  d )  <P  (
b  +P.  c )  <->  ( f  +P.  ( a  +P.  d ) ) 
<P  ( f  +P.  (
b  +P.  c )
) ) )
4238, 40, 16, 41syl3anc 1217 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  d )  <P  ( b  +P.  c
)  <->  ( f  +P.  ( a  +P.  d
) )  <P  (
f  +P.  ( b  +P.  c ) ) ) )
43 addcomprg 7410 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  =  ( s  +P.  r ) )
4443adantl 275 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P. ) )  -> 
( r  +P.  s
)  =  ( s  +P.  r ) )
45 addassprg 7411 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
4645adantl 275 . . . . . . . . . 10  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P.  /\  t  e.  P. ) )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
4716, 15, 19, 44, 46caov12d 5960 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( f  +P.  ( a  +P.  d
) )  =  ( a  +P.  ( f  +P.  d ) ) )
4846, 15, 16, 19caovassd 5938 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  f )  +P.  d )  =  ( a  +P.  ( f  +P.  d ) ) )
4947, 48eqtr4d 2176 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( f  +P.  ( a  +P.  d
) )  =  ( ( a  +P.  f
)  +P.  d )
)
5046, 16, 25, 22caovassd 5938 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
f  +P.  b )  +P.  c )  =  ( f  +P.  ( b  +P.  c ) ) )
5116, 25, 22, 44, 46caov32d 5959 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
f  +P.  b )  +P.  c )  =  ( ( f  +P.  c
)  +P.  b )
)
5250, 51eqtr3d 2175 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( f  +P.  ( b  +P.  c
) )  =  ( ( f  +P.  c
)  +P.  b )
)
5349, 52breq12d 3950 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
f  +P.  ( a  +P.  d ) )  <P 
( f  +P.  (
b  +P.  c )
)  <->  ( ( a  +P.  f )  +P.  d )  <P  (
( f  +P.  c
)  +P.  b )
) )
5442, 53bitrd 187 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  d )  <P  ( b  +P.  c
)  <->  ( ( a  +P.  f )  +P.  d )  <P  (
( f  +P.  c
)  +P.  b )
) )
55 ltaprg 7451 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
5655adantl 275 . . . . . . . 8  |-  ( ( ( ( a  e. 
P.  /\  b  e.  P. )  /\  (
c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  /\  ( r  e.  P.  /\  s  e. 
P.  /\  t  e.  P. ) )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
5756, 18, 30, 19, 44caovord2d 5948 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  f )  <P  ( b  +P.  e
)  <->  ( ( a  +P.  f )  +P.  d )  <P  (
( b  +P.  e
)  +P.  d )
) )
58 addclpr 7369 . . . . . . . . . 10  |-  ( ( e  e.  P.  /\  d  e.  P. )  ->  ( e  +P.  d
)  e.  P. )
5928, 19, 58syl2anc 409 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( e  +P.  d )  e.  P. )
6056, 59, 24, 25, 44caovord2d 5948 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
e  +P.  d )  <P  ( f  +P.  c
)  <->  ( ( e  +P.  d )  +P.  b )  <P  (
( f  +P.  c
)  +P.  b )
) )
6146, 25, 28, 19caovassd 5938 . . . . . . . . . 10  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
b  +P.  e )  +P.  d )  =  ( b  +P.  ( e  +P.  d ) ) )
6244, 25, 59caovcomd 5935 . . . . . . . . . 10  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( b  +P.  ( e  +P.  d
) )  =  ( ( e  +P.  d
)  +P.  b )
)
6361, 62eqtrd 2173 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
b  +P.  e )  +P.  d )  =  ( ( e  +P.  d
)  +P.  b )
)
6463breq1d 3947 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
( b  +P.  e
)  +P.  d )  <P  ( ( f  +P.  c )  +P.  b
)  <->  ( ( e  +P.  d )  +P.  b )  <P  (
( f  +P.  c
)  +P.  b )
) )
6560, 64bitr4d 190 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
e  +P.  d )  <P  ( f  +P.  c
)  <->  ( ( b  +P.  e )  +P.  d )  <P  (
( f  +P.  c
)  +P.  b )
) )
6657, 65orbi12d 783 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
( a  +P.  f
)  <P  ( b  +P.  e )  \/  (
e  +P.  d )  <P  ( f  +P.  c
) )  <->  ( (
( a  +P.  f
)  +P.  d )  <P  ( ( b  +P.  e )  +P.  d
)  \/  ( ( b  +P.  e )  +P.  d )  <P 
( ( f  +P.  c )  +P.  b
) ) ) )
6736, 54, 663imtr4d 202 . . . . 5  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( (
a  +P.  d )  <P  ( b  +P.  c
)  ->  ( (
a  +P.  f )  <P  ( b  +P.  e
)  \/  ( e  +P.  d )  <P 
( f  +P.  c
) ) ) )
68 ltsrprg 7579 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )
)  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  <->  ( a  +P.  d ) 
<P  ( b  +P.  c
) ) )
69683adant3 1002 . . . . 5  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  <->  ( a  +P.  d ) 
<P  ( b  +P.  c
) ) )
70 ltsrprg 7579 . . . . . . 7  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f >. ]  ~R  <->  ( a  +P.  f ) 
<P  ( b  +P.  e
) ) )
71703adant2 1001 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f >. ]  ~R  <->  ( a  +P.  f ) 
<P  ( b  +P.  e
) ) )
72 ltsrprg 7579 . . . . . . . 8  |-  ( ( ( e  e.  P.  /\  f  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )
)  ->  ( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  <->  ( e  +P.  d ) 
<P  ( f  +P.  c
) ) )
7372ancoms 266 . . . . . . 7  |-  ( ( ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  <->  ( e  +P.  d ) 
<P  ( f  +P.  c
) ) )
74733adant1 1000 . . . . . 6  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  <->  ( e  +P.  d ) 
<P  ( f  +P.  c
) ) )
7571, 74orbi12d 783 . . . . 5  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( ( [ <. a ,  b
>. ]  ~R  <R  [ <. e ,  f >. ]  ~R  \/  [ <. e ,  f
>. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) 
<->  ( ( a  +P.  f )  <P  (
b  +P.  e )  \/  ( e  +P.  d
)  <P  ( f  +P.  c ) ) ) )
7667, 69, 753imtr4d 202 . . . 4  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ->  ( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f
>. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d >. ]  ~R  ) ) )
772, 6, 10, 14, 763ecoptocl 6526 . . 3  |-  ( ( x  e.  R.  /\  y  e.  R.  /\  z  e.  R. )  ->  (
x  <R  y  ->  (
x  <R  z  \/  z  <R  y ) ) )
7877rgen3 2522 . 2  |-  A. x  e.  R.  A. y  e. 
R.  A. z  e.  R.  ( x  <R  y  -> 
( x  <R  z  \/  z  <R  y ) )
79 df-iso 4227 . 2  |-  (  <R  Or  R.  <->  (  <R  Po  R.  /\ 
A. x  e.  R.  A. y  e.  R.  A. z  e.  R.  (
x  <R  y  ->  (
x  <R  z  \/  z  <R  y ) ) ) )
801, 78, 79mpbir2an 927 1  |-  <R  Or  R.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   <.cop 3535   class class class wbr 3937    Po wpo 4224    Or wor 4225  (class class class)co 5782   [cec 6435   P.cnp 7123    +P. cpp 7125    <P cltp 7127    ~R cer 7128   R.cnr 7129    <R cltr 7135
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302  df-enr 7558  df-nr 7559  df-ltr 7562
This theorem is referenced by:  1ne0sr  7598  addgt0sr  7607  caucvgsrlemcl  7621  caucvgsrlemfv  7623  suplocsrlemb  7638  suplocsrlempr  7639  suplocsrlem  7640  axpre-ltirr  7714  axpre-ltwlin  7715  axpre-lttrn  7716
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