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Theorem ltsrprg 7519
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 7509 . 2  |-  ~R  e.  _V
2 enrer 7507 . 2  |-  ~R  Er  ( P.  X.  P. )
3 df-nr 7499 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
4 df-ltr 7502 . 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 7508 . . . . 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 7508 . . . . . 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 2117 . . . . . 6  |-  ( ( v  +P.  D )  =  ( u  +P.  C )  <->  ( u  +P.  C )  =  ( v  +P.  D ) )
86, 7syl6bb 195 . . . . 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 578 . . . 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 5749 . . . . . . 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 273 . . . . . 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 510 . . . . . . . . . . 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 508 . . . . . . . . . . 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 7350 . . . . . . . . . . . 12  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( u  +P.  B
)  =  ( B  +P.  u ) )
1514oveq1d 5755 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P. 
C ) )
1612, 13, 15syl2anc 406 . . . . . . . . . 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 511 . . . . . . . . . . 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 7351 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( u  +P.  B
)  +P.  C )  =  ( u  +P.  ( B  +P.  C ) ) )
1912, 13, 17, 18syl3anc 1199 . . . . . . . . . 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 7351 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  u  e.  P.  /\  C  e.  P. )  ->  (
( B  +P.  u
)  +P.  C )  =  ( B  +P.  ( u  +P.  C ) ) )
2113, 12, 17, 20syl3anc 1199 . . . . . . . . . 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 2156 . . . . . . . . 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 5756 . . . . . . . 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 505 . . . . . . . . 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 7309 . . . . . . . . . . . . 13  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
2625ad2ant2lr 499 . . . . . . . . . . . 12  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
27 addclpr 7309 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2827ad2ant2lr 499 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
2926, 28anim12ci 335 . . . . . . . . . . 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 560 . . . . . . . . . 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 111 . . . . . . . . 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 7351 . . . . . . . . 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 1199 . . . . . . . 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 7309 . . . . . . . . . 10  |-  ( ( u  e.  P.  /\  C  e.  P. )  ->  ( u  +P.  C
)  e.  P. )
3512, 17, 34syl2anc 406 . . . . . . . . 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 7351 . . . . . . . . 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 1199 . . . . . . . 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 2158 . . . . . . 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 272 . . . . . 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 509 . . . . . . . . . . . 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 507 . . . . . . . . . . . 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 7350 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  A  e.  P. )  ->  ( v  +P.  A
)  =  ( A  +P.  v ) )
4340, 41, 42syl2anc 406 . . . . . . . . . . 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 5755 . . . . . . . . . 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 512 . . . . . . . . . . 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 7351 . . . . . . . . . . 11  |-  ( ( v  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  (
( v  +P.  A
)  +P.  D )  =  ( v  +P.  ( A  +P.  D
) ) )
4740, 41, 45, 46syl3anc 1199 . . . . . . . . . 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 7351 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  v  e.  P.  /\  D  e.  P. )  ->  (
( A  +P.  v
)  +P.  D )  =  ( A  +P.  ( v  +P.  D
) ) )
4941, 40, 45, 48syl3anc 1199 . . . . . . . . . 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 2156 . . . . . . . . 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 5756 . . . . . . . 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 506 . . . . . . . . 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 7309 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  D  e.  P. )  ->  ( A  +P.  D
)  e.  P. )
5441, 45, 53syl2anc 406 . . . . . . . . 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 7351 . . . . . . . . 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 1199 . . . . . . . 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 7309 . . . . . . . . . 10  |-  ( ( v  e.  P.  /\  D  e.  P. )  ->  ( v  +P.  D
)  e.  P. )
5840, 45, 57syl2anc 406 . . . . . . . . 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 7351 . . . . . . . . 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 1199 . . . . . . . 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 2158 . . . . . . 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 272 . . . . . 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 2158 . . . . 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 114 . . . 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 149 . . 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 7391 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  f  e.  P. )  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
6766adantl 273 . . . 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 7309 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  +P.  u
)  e.  P. )
6924, 12, 68syl2anc 406 . . . 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 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. ) ) )  ->  ( w  +P.  v )  e.  P. )
71 addcomprg 7350 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
7271adantl 273 . . . 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 5907 . . 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 45 . 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 6485 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   <.cop 3498   class class class wbr 3897  (class class class)co 5740   [cec 6393   P.cnp 7063    +P. cpp 7065    <P cltp 7067    ~R cer 7068   R.cnr 7069    <R cltr 7075
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242  df-enr 7498  df-nr 7499  df-ltr 7502
This theorem is referenced by:  gt0srpr  7520  lttrsr  7534  ltposr  7535  ltsosr  7536  0lt1sr  7537  ltasrg  7542  aptisr  7551  mulextsr1  7553  archsr  7554  prsrlt  7559  ltpsrprg  7575  mappsrprg  7576  map2psrprg  7577  pitoregt0  7621
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