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Theorem ltsrprg 7724
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 7714 . 2  |-  ~R  e.  _V
2 enrer 7712 . 2  |-  ~R  Er  ( P.  X.  P. )
3 df-nr 7704 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
4 df-ltr 7707 . 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 7713 . . . . 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 7713 . . . . . 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 2179 . . . . . 6  |-  ( ( v  +P.  D )  =  ( u  +P.  C )  <->  ( u  +P.  C )  =  ( v  +P.  D ) )
86, 7bitrdi 196 . . . . 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 606 . . . 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 5877 . . . . . . 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 277 . . . . . 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 538 . . . . . . . . . . 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 536 . . . . . . . . . . 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 7555 . . . . . . . . . . . 12  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( u  +P.  B
)  =  ( B  +P.  u ) )
1514oveq1d 5883 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P. 
C ) )
1612, 13, 15syl2anc 411 . . . . . . . . . 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 539 . . . . . . . . . . 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 7556 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( u  +P.  B
)  +P.  C )  =  ( u  +P.  ( B  +P.  C ) ) )
1912, 13, 17, 18syl3anc 1238 . . . . . . . . . 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 7556 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  u  e.  P.  /\  C  e.  P. )  ->  (
( B  +P.  u
)  +P.  C )  =  ( B  +P.  ( u  +P.  C ) ) )
2113, 12, 17, 20syl3anc 1238 . . . . . . . . . 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 2218 . . . . . . . . 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 5884 . . . . . . . 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 533 . . . . . . . . 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 7514 . . . . . . . . . . . . 13  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
2625ad2ant2lr 510 . . . . . . . . . . . 12  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
27 addclpr 7514 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
2827ad2ant2lr 510 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
2926, 28anim12ci 339 . . . . . . . . . . 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 588 . . . . . . . . . 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 112 . . . . . . . . 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 7556 . . . . . . . . 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 1238 . . . . . . . 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 7514 . . . . . . . . . 10  |-  ( ( u  e.  P.  /\  C  e.  P. )  ->  ( u  +P.  C
)  e.  P. )
3512, 17, 34syl2anc 411 . . . . . . . . 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 7556 . . . . . . . . 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 1238 . . . . . . . 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 2220 . . . . . . 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 276 . . . . . 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 537 . . . . . . . . . . . 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 535 . . . . . . . . . . . 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 7555 . . . . . . . . . . . 12  |-  ( ( v  e.  P.  /\  A  e.  P. )  ->  ( v  +P.  A
)  =  ( A  +P.  v ) )
4340, 41, 42syl2anc 411 . . . . . . . . . . 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 5883 . . . . . . . . . 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 540 . . . . . . . . . . 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 7556 . . . . . . . . . . 11  |-  ( ( v  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  (
( v  +P.  A
)  +P.  D )  =  ( v  +P.  ( A  +P.  D
) ) )
4740, 41, 45, 46syl3anc 1238 . . . . . . . . . 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 7556 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  v  e.  P.  /\  D  e.  P. )  ->  (
( A  +P.  v
)  +P.  D )  =  ( A  +P.  ( v  +P.  D
) ) )
4941, 40, 45, 48syl3anc 1238 . . . . . . . . . 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 2218 . . . . . . . . 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 5884 . . . . . . . 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 534 . . . . . . . . 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 7514 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  D  e.  P. )  ->  ( A  +P.  D
)  e.  P. )
5441, 45, 53syl2anc 411 . . . . . . . . 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 7556 . . . . . . . . 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 1238 . . . . . . . 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 7514 . . . . . . . . . 10  |-  ( ( v  e.  P.  /\  D  e.  P. )  ->  ( v  +P.  D
)  e.  P. )
5840, 45, 57syl2anc 411 . . . . . . . . 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 7556 . . . . . . . . 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 1238 . . . . . . . 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 2220 . . . . . . 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 276 . . . . . 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 2220 . . . . 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 115 . . . 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 150 . . 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 7596 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P.  /\  f  e.  P. )  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
6766adantl 277 . . . 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 7514 . . . . 5  |-  ( ( z  e.  P.  /\  u  e.  P. )  ->  ( z  +P.  u
)  e.  P. )
6924, 12, 68syl2anc 411 . . . 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 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. ) ) )  ->  ( w  +P.  v )  e.  P. )
71 addcomprg 7555 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  y
)  =  ( y  +P.  x ) )
7271adantl 277 . . . 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 6038 . . 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 6618 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   <.cop 3594   class class class wbr 4000  (class class class)co 5868   [cec 6526   P.cnp 7268    +P. cpp 7270    <P cltp 7272    ~R cer 7273   R.cnr 7274    <R cltr 7280
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-2o 6411  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329  df-ltnqqs 7330  df-enq0 7401  df-nq0 7402  df-0nq0 7403  df-plq0 7404  df-mq0 7405  df-inp 7443  df-iplp 7445  df-iltp 7447  df-enr 7703  df-nr 7704  df-ltr 7707
This theorem is referenced by:  gt0srpr  7725  lttrsr  7739  ltposr  7740  ltsosr  7741  0lt1sr  7742  ltasrg  7747  aptisr  7756  mulextsr1  7758  archsr  7759  prsrlt  7764  ltpsrprg  7780  mappsrprg  7781  map2psrprg  7782  pitoregt0  7826
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