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Theorem ltsosr 7765
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 7764 . 2  |-  <R  Po  R.
2 df-nr 7728 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
3 breq1 4008 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  x  <R  [
<. c ,  d >. ]  ~R  ) )
4 breq1 4008 . . . . . 6  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f
>. ]  ~R  <->  x  <R  [
<. e ,  f >. ]  ~R  ) )
54orbi1d 791 . . . . 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 234 . . . 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 4009 . . . . 5  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( x  <R  [ <. c ,  d >. ]  ~R  <->  x 
<R  y ) )
8 breq2 4009 . . . . . 6  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  [ <. e ,  f >. ]  ~R  <R  y ) )
98orbi2d 790 . . . . 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 234 . . . 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 4009 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( x  <R  [ <. e ,  f >. ]  ~R  <->  x 
<R  z ) )
12 breq1 4008 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( [ <. e ,  f >. ]  ~R  <R  y  <->  z  <R  y
) )
1311, 12orbi12d 793 . . . . 5  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( ( x  <R  [
<. e ,  f >. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y )  <->  ( x  <R  z  \/  z  <R 
y ) ) )
1413imbi2d 230 . . . 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 1021 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  a  e.  P. )
16 simp3r 1026 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  f  e.  P. )
17 addclpr 7538 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  f  e.  P. )  ->  ( a  +P.  f
)  e.  P. )
1815, 16, 17syl2anc 411 . . . . . . . 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 1024 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  d  e.  P. )
20 addclpr 7538 . . . . . . . 8  |-  ( ( ( a  +P.  f
)  e.  P.  /\  d  e.  P. )  ->  ( ( a  +P.  f )  +P.  d
)  e.  P. )
2118, 19, 20syl2anc 411 . . . . . . 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 1023 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  c  e.  P. )
23 addclpr 7538 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  c  e.  P. )  ->  ( f  +P.  c
)  e.  P. )
2416, 22, 23syl2anc 411 . . . . . . . 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 1022 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  b  e.  P. )
26 addclpr 7538 . . . . . . . 8  |-  ( ( ( f  +P.  c
)  e.  P.  /\  b  e.  P. )  ->  ( ( f  +P.  c )  +P.  b
)  e.  P. )
2724, 25, 26syl2anc 411 . . . . . . 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 1025 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  e  e.  P. )
29 addclpr 7538 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  e  e.  P. )  ->  ( b  +P.  e
)  e.  P. )
3025, 28, 29syl2anc 411 . . . . . . . 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 7538 . . . . . . . 8  |-  ( ( ( b  +P.  e
)  e.  P.  /\  d  e.  P. )  ->  ( ( b  +P.  e )  +P.  d
)  e.  P. )
3230, 19, 31syl2anc 411 . . . . . . 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 7597 . . . . . . . 8  |-  <P  Or  P.
34 sowlin 4322 . . . . . . . 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 424 . . . . . . 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 1238 . . . . . 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 7538 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  d  e.  P. )  ->  ( a  +P.  d
)  e.  P. )
3815, 19, 37syl2anc 411 . . . . . . . 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 7538 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  c  e.  P. )  ->  ( b  +P.  c
)  e.  P. )
4025, 22, 39syl2anc 411 . . . . . . . 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 7620 . . . . . . . 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 1238 . . . . . . 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 7579 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  =  ( s  +P.  r ) )
4443adantl 277 . . . . . . . . . 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 7580 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
4645adantl 277 . . . . . . . . . 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 6058 . . . . . . . . 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 6036 . . . . . . . . 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 2213 . . . . . . . 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 6036 . . . . . . . . 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 6057 . . . . . . . . 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 2212 . . . . . . . 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 4018 . . . . . . 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 188 . . . . . 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 7620 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
5655adantl 277 . . . . . . . 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 6046 . . . . . . 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 7538 . . . . . . . . . 10  |-  ( ( e  e.  P.  /\  d  e.  P. )  ->  ( e  +P.  d
)  e.  P. )
5928, 19, 58syl2anc 411 . . . . . . . . 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 6046 . . . . . . . 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 6036 . . . . . . . . . 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 6033 . . . . . . . . . 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 2210 . . . . . . . . 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 4015 . . . . . . . 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 191 . . . . . . 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 793 . . . . . 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 203 . . . . 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 7748 . . . . . 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 1017 . . . . 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 7748 . . . . . . 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 1016 . . . . . 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 7748 . . . . . . . 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 268 . . . . . . 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 1015 . . . . . 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 793 . . . . 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 203 . . . 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 6626 . . 3  |-  ( ( x  e.  R.  /\  y  e.  R.  /\  z  e.  R. )  ->  (
x  <R  y  ->  (
x  <R  z  \/  z  <R  y ) ) )
7877rgen3 2564 . 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 4299 . 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 942 1  |-  <R  Or  R.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   <.cop 3597   class class class wbr 4005    Po wpo 4296    Or wor 4297  (class class class)co 5877   [cec 6535   P.cnp 7292    +P. cpp 7294    <P cltp 7296    ~R cer 7297   R.cnr 7298    <R cltr 7304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471  df-enr 7727  df-nr 7728  df-ltr 7731
This theorem is referenced by:  1ne0sr  7767  addgt0sr  7776  caucvgsrlemcl  7790  caucvgsrlemfv  7792  suplocsrlemb  7807  suplocsrlempr  7808  suplocsrlem  7809  axpre-ltirr  7883  axpre-ltwlin  7884  axpre-lttrn  7885
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