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

Theorem ltsrprg 7777
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 7767 . 2  |-  ~R  e.  _V
2 enrer 7765 . 2  |-  ~R  Er  ( P.  X.  P. )
3 df-nr 7757 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
4 df-ltr 7760 . 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 7766 . . . . 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 7766 . . . . . 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 2191 . . . . . 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 5906 . . . . . . 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 7608 . . . . . . . . . . . 12  |-  ( ( u  e.  P.  /\  B  e.  P. )  ->  ( u  +P.  B
)  =  ( B  +P.  u ) )
1514oveq1d 5912 . . . . . . . . . . 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 7609 . . . . . . . . . . 11  |-  ( ( u  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( u  +P.  B
)  +P.  C )  =  ( u  +P.  ( B  +P.  C ) ) )
1912, 13, 17, 18syl3anc 1249 . . . . . . . . . 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 7609 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  u  e.  P.  /\  C  e.  P. )  ->  (
( B  +P.  u
)  +P.  C )  =  ( B  +P.  ( u  +P.  C ) ) )
2113, 12, 17, 20syl3anc 1249 . . . . . . . . . 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 2230 . . . . . . . . 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 5913 . . . . . . . 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 7567 . . . . . . . . . . . . 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 7567 . . . . . . . . . . . . 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 7609 . . . . . . . . 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 1249 . . . . . . . 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 7567 . . . . . . . . . 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 7609 . . . . . . . . 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 1249 . . . . . . . 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 2232 . . . . . . 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 7608 . . . . . . . . . . . 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 5912 . . . . . . . . . 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 7609 . . . . . . . . . . 11  |-  ( ( v  e.  P.  /\  A  e.  P.  /\  D  e.  P. )  ->  (
( v  +P.  A
)  +P.  D )  =  ( v  +P.  ( A  +P.  D
) ) )
4740, 41, 45, 46syl3anc 1249 . . . . . . . . . 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 7609 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  v  e.  P.  /\  D  e.  P. )  ->  (
( A  +P.  v
)  +P.  D )  =  ( A  +P.  ( v  +P.  D
) ) )
4941, 40, 45, 48syl3anc 1249 . . . . . . . . . 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 2230 . . . . . . . . 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 5913 . . . . . . . 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 7567 . . . . . . . . . 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 7609 . . . . . . . . 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 1249 . . . . . . . 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 7567 . . . . . . . . . 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 7609 . . . . . . . . 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 1249 . . . . . . . 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 2232 . . . . . . 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 2232 . . . . 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 7649 . . . . 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 7567 . . . . 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 7608 . . . . 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 6068 . . 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 6652 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 980    = wceq 1364    e. wcel 2160   <.cop 3610   class class class wbr 4018  (class class class)co 5897   [cec 6558   P.cnp 7321    +P. cpp 7323    <P cltp 7325    ~R cer 7326   R.cnr 7327    <R cltr 7333
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-iplp 7498  df-iltp 7500  df-enr 7756  df-nr 7757  df-ltr 7760
This theorem is referenced by:  gt0srpr  7778  lttrsr  7792  ltposr  7793  ltsosr  7794  0lt1sr  7795  ltasrg  7800  aptisr  7809  mulextsr1  7811  archsr  7812  prsrlt  7817  ltpsrprg  7833  mappsrprg  7834  map2psrprg  7835  pitoregt0  7879
  Copyright terms: Public domain W3C validator