ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltsosr Unicode version

Theorem ltsosr 7003
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 7002 . 2  |-  <R  Po  R.
2 df-nr 6966 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
3 breq1 3796 . . . . 5  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  x  <R  [
<. c ,  d >. ]  ~R  ) )
4 breq1 3796 . . . . . 6  |-  ( [
<. a ,  b >. ]  ~R  =  x  -> 
( [ <. a ,  b >. ]  ~R  <R  [ <. e ,  f
>. ]  ~R  <->  x  <R  [
<. e ,  f >. ]  ~R  ) )
54orbi1d 738 . . . . 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 232 . . . 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 3797 . . . . 5  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( x  <R  [ <. c ,  d >. ]  ~R  <->  x 
<R  y ) )
8 breq2 3797 . . . . . 6  |-  ( [
<. c ,  d >. ]  ~R  =  y  -> 
( [ <. e ,  f >. ]  ~R  <R  [ <. c ,  d
>. ]  ~R  <->  [ <. e ,  f >. ]  ~R  <R  y ) )
98orbi2d 737 . . . . 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 232 . . . 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 3797 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( x  <R  [ <. e ,  f >. ]  ~R  <->  x 
<R  z ) )
12 breq1 3796 . . . . . 6  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( [ <. e ,  f >. ]  ~R  <R  y  <->  z  <R  y
) )
1311, 12orbi12d 740 . . . . 5  |-  ( [
<. e ,  f >. ]  ~R  =  z  -> 
( ( x  <R  [
<. e ,  f >. ]  ~R  \/  [ <. e ,  f >. ]  ~R  <R  y )  <->  ( x  <R  z  \/  z  <R 
y ) ) )
1413imbi2d 228 . . . 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 963 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  a  e.  P. )
16 simp3r 968 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  f  e.  P. )
17 addclpr 6789 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  f  e.  P. )  ->  ( a  +P.  f
)  e.  P. )
1815, 16, 17syl2anc 403 . . . . . . . 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 966 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  d  e.  P. )
20 addclpr 6789 . . . . . . . 8  |-  ( ( ( a  +P.  f
)  e.  P.  /\  d  e.  P. )  ->  ( ( a  +P.  f )  +P.  d
)  e.  P. )
2118, 19, 20syl2anc 403 . . . . . . 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 965 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  c  e.  P. )
23 addclpr 6789 . . . . . . . . 9  |-  ( ( f  e.  P.  /\  c  e.  P. )  ->  ( f  +P.  c
)  e.  P. )
2416, 22, 23syl2anc 403 . . . . . . . 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 964 . . . . . . . 8  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  b  e.  P. )
26 addclpr 6789 . . . . . . . 8  |-  ( ( ( f  +P.  c
)  e.  P.  /\  b  e.  P. )  ->  ( ( f  +P.  c )  +P.  b
)  e.  P. )
2724, 25, 26syl2anc 403 . . . . . . 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 967 . . . . . . . . 9  |-  ( ( ( a  e.  P.  /\  b  e.  P. )  /\  ( c  e.  P.  /\  d  e.  P. )  /\  ( e  e.  P.  /\  f  e.  P. )
)  ->  e  e.  P. )
29 addclpr 6789 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  e  e.  P. )  ->  ( b  +P.  e
)  e.  P. )
3025, 28, 29syl2anc 403 . . . . . . . 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 6789 . . . . . . . 8  |-  ( ( ( b  +P.  e
)  e.  P.  /\  d  e.  P. )  ->  ( ( b  +P.  e )  +P.  d
)  e.  P. )
3230, 19, 31syl2anc 403 . . . . . . 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 6848 . . . . . . . 8  |-  <P  Or  P.
34 sowlin 4083 . . . . . . . 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 415 . . . . . . 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 1170 . . . . . 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 6789 . . . . . . . . 9  |-  ( ( a  e.  P.  /\  d  e.  P. )  ->  ( a  +P.  d
)  e.  P. )
3815, 19, 37syl2anc 403 . . . . . . . 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 6789 . . . . . . . . 9  |-  ( ( b  e.  P.  /\  c  e.  P. )  ->  ( b  +P.  c
)  e.  P. )
4025, 22, 39syl2anc 403 . . . . . . . 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 6871 . . . . . . . 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 1170 . . . . . . 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 6830 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P. )  ->  ( r  +P.  s
)  =  ( s  +P.  r ) )
4443adantl 271 . . . . . . . . . 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 6831 . . . . . . . . . . 11  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
( r  +P.  s
)  +P.  t )  =  ( r  +P.  ( s  +P.  t
) ) )
4645adantl 271 . . . . . . . . . 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 5713 . . . . . . . . 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 5691 . . . . . . . . 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 2117 . . . . . . . 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 5691 . . . . . . . . 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 5712 . . . . . . . . 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 2116 . . . . . . . 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 3806 . . . . . . 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 186 . . . . . 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 6871 . . . . . . . . 9  |-  ( ( r  e.  P.  /\  s  e.  P.  /\  t  e.  P. )  ->  (
r  <P  s  <->  ( t  +P.  r )  <P  (
t  +P.  s )
) )
5655adantl 271 . . . . . . . 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 5701 . . . . . . 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 6789 . . . . . . . . . 10  |-  ( ( e  e.  P.  /\  d  e.  P. )  ->  ( e  +P.  d
)  e.  P. )
5928, 19, 58syl2anc 403 . . . . . . . . 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 5701 . . . . . . . 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 5691 . . . . . . . . . 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 5688 . . . . . . . . . 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 2114 . . . . . . . . 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 3803 . . . . . . . 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 189 . . . . . . 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 740 . . . . . 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 201 . . . . 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 6986 . . . . . 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 959 . . . . 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 6986 . . . . . . 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 958 . . . . . 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 6986 . . . . . . . 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 264 . . . . . . 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 957 . . . . . 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 740 . . . . 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 201 . . . 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 6261 . . 3  |-  ( ( x  e.  R.  /\  y  e.  R.  /\  z  e.  R. )  ->  (
x  <R  y  ->  (
x  <R  z  \/  z  <R  y ) ) )
7877rgen3 2449 . 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 4060 . 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 884 1  |-  <R  Or  R.
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2349   <.cop 3409   class class class wbr 3793    Po wpo 4057    Or wor 4058  (class class class)co 5543   [cec 6170   P.cnp 6543    +P. cpp 6545    <P cltp 6547    ~R cer 6548   R.cnr 6549    <R cltr 6555
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-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  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-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722  df-enr 6965  df-nr 6966  df-ltr 6969
This theorem is referenced by:  1ne0sr  7005  addgt0sr  7014  caucvgsrlemcl  7027  caucvgsrlemfv  7029  axpre-ltirr  7110  axpre-ltwlin  7111  axpre-lttrn  7112
  Copyright terms: Public domain W3C validator