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

Theorem ltsrprg 6986
Description: Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
Assertion
Ref Expression
ltsrprg  |-  ( ( ( 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 ) ) )

Proof of Theorem ltsrprg
Dummy variables  x  y  z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrex 6976 . 2  |-  ~R  e.  _V
2 enrer 6974 . 2  |-  ~R  Er  ( P.  X.  P. )
3 df-nr 6966 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
4 df-ltr 6969 . 2  |-  <R  =  { <. x ,  y
>.  |  ( (
x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u
)  <P  ( w  +P.  v ) ) ) }
5 enreceq 6975 . . . . 5  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  <->  ( z  +P.  B )  =  ( w  +P.  A ) ) )
6 enreceq 6975 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  <->  ( v  +P.  D )  =  ( u  +P.  C ) ) )
7 eqcom 2084 . . . . . 6  |-  ( ( v  +P.  D )  =  ( u  +P.  C )  <->  ( u  +P.  C )  =  ( v  +P.  D ) )
86, 7syl6bb 194 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  <->  ( u  +P.  C )  =  ( v  +P. 
D ) ) )
95, 8bi2anan9 571 . . . 4  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( [
<. z ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  <->  ( (
z  +P.  B )  =  ( w  +P.  A )  /\  ( u  +P.  C )  =  ( v  +P.  D
) ) ) )
10 oveq12 5552 . . . . . . 7  |-  ( ( ( z  +P.  B
)  =  ( w  +P.  A )  /\  ( u  +P.  C )  =  ( v  +P. 
D ) )  -> 
( ( z  +P. 
B )  +P.  (
u  +P.  C )
)  =  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) ) )
1110adantl 271 . . . . . 6  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( ( z  +P.  B )  =  ( w  +P.  A
)  /\  ( u  +P.  C )  =  ( v  +P.  D ) ) )  ->  (
( z  +P.  B
)  +P.  ( u  +P.  C ) )  =  ( ( w  +P.  A )  +P.  ( v  +P.  D ) ) )
12 simprlr 505 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  u  e.  P. )
13 simplrr 503 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  B  e.  P. )
14 addcomprg 6830 . . . . . . . . . . . 12  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( u  +P.  B
)  =  ( B  +P.  u ) )
1514oveq1d 5558 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P. 
C ) )
1612, 13, 15syl2anc 403 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( u  +P.  B )  +P. 
C )  =  ( ( B  +P.  u
)  +P.  C )
)
17 simprrl 506 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  C  e.  P. )
18 addassprg 6831 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( u  +P.  B
)  +P.  C )  =  ( u  +P.  ( B  +P.  C ) ) )
1912, 13, 17, 18syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( u  +P.  B )  +P. 
C )  =  ( u  +P.  ( B  +P.  C ) ) )
20 addassprg 6831 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  u  e.  P.  /\  C  e.  P. )  ->  (
( B  +P.  u
)  +P.  C )  =  ( B  +P.  ( u  +P.  C ) ) )
2113, 12, 17, 20syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( B  +P.  u )  +P. 
C )  =  ( B  +P.  ( u  +P.  C ) ) )
2216, 19, 213eqtr3d 2122 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) ) )
2322oveq2d 5559 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) ) )
24 simplll 500 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  z  e.  P. )
25 addclpr 6789 . . . . . . . . . . . . 13  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
2625ad2ant2lr 494 . . . . . . . . . . . 12  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
27 addclpr 6789 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2827ad2ant2lr 494 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
2926, 28anim12ci 332 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  (
v  e.  P.  /\  u  e.  P. )
)  /\  ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( B  +P.  C )  e. 
P.  /\  ( w  +P.  v )  e.  P. ) )
3029an4s 553 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( B  +P.  C )  e. 
P.  /\  ( w  +P.  v )  e.  P. ) )
3130simpld 110 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( B  +P.  C )  e.  P. )
32 addassprg 6831 . . . . . . . . 9  |-  ( ( z  e.  P.  /\  u  e.  P.  /\  ( B  +P.  C )  e. 
P. )  ->  (
( z  +P.  u
)  +P.  ( B  +P.  C ) )  =  ( z  +P.  (
u  +P.  ( B  +P.  C ) ) ) )
3324, 12, 31, 32syl3anc 1170 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( z  +P.  u )  +P.  ( B  +P.  C
) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) ) )
34 addclpr 6789 . . . . . . . . . 10  |-  ( ( u  e.  P.  /\  C  e.  P. )  ->  ( u  +P.  C
)  e.  P. )
3512, 17, 34syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( u  +P.  C )  e.  P. )
36 addassprg 6831 . . . . . . . . 9  |-  ( ( z  e.  P.  /\  B  e.  P.  /\  (
u  +P.  C )  e.  P. )  ->  (
( z  +P.  B
)  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) ) )
3724, 13, 35, 36syl3anc 1170 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( z  +P.  B )  +P.  ( u  +P.  C
) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) ) )
3823, 33, 373eqtr4d 2124 . . . . . . 7  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( z  +P.  u )  +P.  ( B  +P.  C
) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) ) )
3938adantr 270 . . . . . 6  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( ( z  +P.  B )  =  ( w  +P.  A
)  /\  ( u  +P.  C )  =  ( v  +P.  D ) ) )  ->  (
( z  +P.  u
)  +P.  ( B  +P.  C ) )  =  ( ( z  +P. 
B )  +P.  (
u  +P.  C )
) )
40 simprll 504 . . . . . . . . . . . 12  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  v  e.  P. )
41 simplrl 502 . . . . . . . . . . . 12  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  A  e.  P. )
42 addcomprg 6830 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  A  e.  P. )  ->  ( v  +P.  A
)  =  ( A  +P.  v ) )
4340, 41, 42syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( v  +P. 
A )  =  ( A  +P.  v ) )
4443oveq1d 5558 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( v  +P.  A )  +P. 
D )  =  ( ( A  +P.  v
)  +P.  D )
)
45 simprrr 507 . . . . . . . . . . 11  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  D  e.  P. )
46 addassprg 6831 . . . . . . . . . . 11  |-  ( ( v  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  (
( v  +P.  A
)  +P.  D )  =  ( v  +P.  ( A  +P.  D
) ) )
4740, 41, 45, 46syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( v  +P.  A )  +P. 
D )  =  ( v  +P.  ( A  +P.  D ) ) )
48 addassprg 6831 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  v  e.  P.  /\  D  e.  P. )  ->  (
( A  +P.  v
)  +P.  D )  =  ( A  +P.  ( v  +P.  D
) ) )
4941, 40, 45, 48syl3anc 1170 . . . . . . . . . 10  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( A  +P.  v )  +P. 
D )  =  ( A  +P.  ( v  +P.  D ) ) )
5044, 47, 493eqtr3d 2122 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( v  +P.  ( A  +P.  D
) )  =  ( A  +P.  ( v  +P.  D ) ) )
5150oveq2d 5559 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )  =  ( w  +P.  ( A  +P.  ( v  +P.  D
) ) ) )
52 simpllr 501 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  w  e.  P. )
53 addclpr 6789 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  D  e.  P. )  ->  ( A  +P.  D
)  e.  P. )
5441, 45, 53syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( A  +P.  D )  e.  P. )
55 addassprg 6831 . . . . . . . . 9  |-  ( ( w  e.  P.  /\  v  e.  P.  /\  ( A  +P.  D )  e. 
P. )  ->  (
( w  +P.  v
)  +P.  ( A  +P.  D ) )  =  ( w  +P.  (
v  +P.  ( A  +P.  D ) ) ) )
5652, 40, 54, 55syl3anc 1170 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( w  +P.  v )  +P.  ( A  +P.  D
) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) ) )
57 addclpr 6789 . . . . . . . . . 10  |-  ( ( v  e.  P.  /\  D  e.  P. )  ->  ( v  +P.  D
)  e.  P. )
5840, 45, 57syl2anc 403 . . . . . . . . 9  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( v  +P. 
D )  e.  P. )
59 addassprg 6831 . . . . . . . . 9  |-  ( ( w  e.  P.  /\  A  e.  P.  /\  (
v  +P.  D )  e.  P. )  ->  (
( w  +P.  A
)  +P.  ( v  +P.  D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) ) )
6052, 41, 58, 59syl3anc 1170 . . . . . . . 8  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( w  +P.  A )  +P.  ( v  +P.  D
) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) ) )
6151, 56, 603eqtr4d 2124 . . . . . . 7  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( w  +P.  v )  +P.  ( A  +P.  D
) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) ) )
6261adantr 270 . . . . . 6  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( ( z  +P.  B )  =  ( w  +P.  A
)  /\  ( u  +P.  C )  =  ( v  +P.  D ) ) )  ->  (
( w  +P.  v
)  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A )  +P.  ( v  +P.  D ) ) )
6311, 39, 623eqtr4d 2124 . . . . 5  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( ( z  +P.  B )  =  ( w  +P.  A
)  /\  ( u  +P.  C )  =  ( v  +P.  D ) ) )  ->  (
( z  +P.  u
)  +P.  ( B  +P.  C ) )  =  ( ( w  +P.  v )  +P.  ( A  +P.  D ) ) )
6463ex 113 . . . 4  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( ( z  +P.  B )  =  ( w  +P.  A )  /\  ( u  +P.  C )  =  ( v  +P.  D
) )  ->  (
( z  +P.  u
)  +P.  ( B  +P.  C ) )  =  ( ( w  +P.  v )  +P.  ( A  +P.  D ) ) ) )
659, 64sylbid 148 . . 3  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( [
<. z ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  ->  (
( z  +P.  u
)  +P.  ( B  +P.  C ) )  =  ( ( w  +P.  v )  +P.  ( A  +P.  D ) ) ) )
66 ltaprg 6871 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  f  e.  P. )  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
6766adantl 271 . . . 4  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( x  e. 
P.  /\  y  e.  P.  /\  f  e.  P. ) )  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
68 addclpr 6789 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  +P.  u
)  e.  P. )
6924, 12, 68syl2anc 403 . . . 4  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( z  +P.  u )  e.  P. )
7030simprd 112 . . . 4  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( w  +P.  v )  e.  P. )
71 addcomprg 6830 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
7271adantl 271 . . . 4  |-  ( ( ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  /\  ( x  e. 
P.  /\  y  e.  P. ) )  ->  (
x  +P.  y )  =  ( y  +P.  x ) )
7367, 69, 31, 70, 72, 54caovord3d 5702 . . 3  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( w  +P.  v
)  +P.  ( A  +P.  D ) )  -> 
( ( z  +P.  u )  <P  (
w  +P.  v )  <->  ( A  +P.  D ) 
<P  ( B  +P.  C
) ) ) )
7465, 73syld 44 . 2  |-  ( ( ( ( z  e. 
P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. ) )  /\  ( ( v  e. 
P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) ) )  ->  ( ( [
<. z ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  ->  (
( z  +P.  u
)  <P  ( w  +P.  v )  <->  ( A  +P.  D )  <P  ( B  +P.  C ) ) ) )
751, 2, 3, 4, 74brecop 6262 1  |-  ( ( ( 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 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3409   class class class wbr 3793  (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:  gt0srpr  6987  lttrsr  7001  ltposr  7002  ltsosr  7003  0lt1sr  7004  ltasrg  7009  aptisr  7017  mulextsr1  7019  archsr  7020  prsrlt  7025  pitoregt0  7079
  Copyright terms: Public domain W3C validator