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Theorem ltsrpr 9506
Description: Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
ltsrpr  |-  ( [
<. A ,  B >. ]  ~R  <R  [ <. C ,  D >. ]  ~R  <->  ( A  +P.  D )  <P  ( B  +P.  C ) )

Proof of Theorem ltsrpr
Dummy variables  x  y  z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrex 9496 . 2  |-  ~R  e.  _V
2 enrer 9494 . . 3  |-  ~R  Er  ( P.  X.  P. )
3 erdm 7378 . . 3  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
42, 3ax-mp 5 . 2  |-  dom  ~R  =  ( P.  X.  P. )
5 df-nr 9486 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6 ltrelsr 9497 . 2  |-  <R  C_  ( R.  X.  R. )
7 ltrelpr 9428 . 2  |-  <P  C_  ( P.  X.  P. )
8 0npr 9422 . 2  |-  -.  (/)  e.  P.
9 dmplp 9442 . 2  |-  dom  +P.  =  ( P.  X.  P. )
10 df-ltr 9489 . . 3  |-  <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 ) ) ) }
11 addclpr 9448 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
1211ad2ant2lr 755 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
13 addclpr 9448 . . . . . . 7  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
1413ad2ant2lr 755 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
1512, 14anim12ci 571 . . . . 5  |-  ( ( ( ( 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. ) )
1615an4s 836 . . . 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. ) ) )  ->  ( ( B  +P.  C )  e. 
P.  /\  ( w  +P.  v )  e.  P. ) )
17 enreceq 9495 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( A  e.  P.  /\  B  e.  P. )
)  ->  ( [ <. z ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  <->  ( z  +P.  B )  =  ( w  +P.  A ) ) )
18 enreceq 9495 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  <->  ( v  +P.  D )  =  ( u  +P.  C ) ) )
19 eqcom 2460 . . . . . . 7  |-  ( ( v  +P.  D )  =  ( u  +P.  C )  <->  ( u  +P.  C )  =  ( v  +P.  D ) )
2018, 19syl6bb 265 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  <->  ( u  +P.  C )  =  ( v  +P. 
D ) ) )
2117, 20bi2anan9 885 . . . . 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 ,  w >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. v ,  u >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  <->  ( (
z  +P.  B )  =  ( w  +P.  A )  /\  ( u  +P.  C )  =  ( v  +P.  D
) ) ) )
22 oveq12 6304 . . . . . 6  |-  ( ( ( 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 ) ) )
23 addcompr 9451 . . . . . . . . . 10  |-  ( u  +P.  B )  =  ( B  +P.  u
)
2423oveq1i 6305 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P.  C
)
25 addasspr 9452 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( u  +P.  ( B  +P.  C ) )
26 addasspr 9452 . . . . . . . . 9  |-  ( ( B  +P.  u )  +P.  C )  =  ( B  +P.  (
u  +P.  C )
)
2724, 25, 263eqtr3i 2483 . . . . . . . 8  |-  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) )
2827oveq2i 6306 . . . . . . 7  |-  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
29 addasspr 9452 . . . . . . 7  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )
30 addasspr 9452 . . . . . . 7  |-  ( ( z  +P.  B )  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
3128, 29, 303eqtr4i 2485 . . . . . 6  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) )
32 addcompr 9451 . . . . . . . . . 10  |-  ( v  +P.  A )  =  ( A  +P.  v
)
3332oveq1i 6305 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( ( A  +P.  v )  +P.  D
)
34 addasspr 9452 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( v  +P.  ( A  +P.  D ) )
35 addasspr 9452 . . . . . . . . 9  |-  ( ( A  +P.  v )  +P.  D )  =  ( A  +P.  (
v  +P.  D )
)
3633, 34, 353eqtr3i 2483 . . . . . . . 8  |-  ( v  +P.  ( A  +P.  D ) )  =  ( A  +P.  ( v  +P.  D ) )
3736oveq2i 6306 . . . . . . 7  |-  ( w  +P.  ( v  +P.  ( A  +P.  D
) ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
38 addasspr 9452 . . . . . . 7  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )
39 addasspr 9452 . . . . . . 7  |-  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
4037, 38, 393eqtr4i 2485 . . . . . 6  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) )
4122, 31, 403eqtr4g 2512 . . . . 5  |-  ( ( ( 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
) ) )
4221, 41syl6bi 232 . . . 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.  u
)  +P.  ( B  +P.  C ) )  =  ( ( w  +P.  v )  +P.  ( A  +P.  D ) ) ) )
43 ovex 6323 . . . . 5  |-  ( z  +P.  u )  e. 
_V
44 ovex 6323 . . . . 5  |-  ( B  +P.  C )  e. 
_V
45 ltapr 9475 . . . . 5  |-  ( f  e.  P.  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
46 ovex 6323 . . . . 5  |-  ( w  +P.  v )  e. 
_V
47 addcompr 9451 . . . . 5  |-  ( x  +P.  y )  =  ( y  +P.  x
)
48 ovex 6323 . . . . 5  |-  ( A  +P.  D )  e. 
_V
4943, 44, 45, 46, 47, 48caovord3 6487 . . . 4  |-  ( ( ( ( B  +P.  C )  e.  P.  /\  ( w  +P.  v )  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
) ) )
5016, 42, 49syl6an 548 . . 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  ( w  +P.  v )  <->  ( A  +P.  D )  <P  ( B  +P.  C ) ) ) )
511, 2, 5, 10, 50brecop 7461 . 2  |-  ( ( ( 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 ) ) )
521, 4, 5, 6, 7, 8, 9, 51brecop2 7462 1  |-  ( [
<. A ,  B >. ]  ~R  <R  [ <. C ,  D >. ]  ~R  <->  ( A  +P.  D )  <P  ( B  +P.  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889   <.cop 3976   class class class wbr 4405    X. cxp 4835   dom cdm 4837  (class class class)co 6295    Er wer 7365   [cec 7366   P.cnp 9289    +P. cpp 9291    <P cltp 9293    ~R cer 9294   R.cnr 9295    <R cltr 9301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-omul 7192  df-er 7368  df-ec 7370  df-qs 7374  df-ni 9302  df-pli 9303  df-mi 9304  df-lti 9305  df-plpq 9338  df-mpq 9339  df-ltpq 9340  df-enq 9341  df-nq 9342  df-erq 9343  df-plq 9344  df-mq 9345  df-1nq 9346  df-rq 9347  df-ltnq 9348  df-np 9411  df-plp 9413  df-ltp 9415  df-enr 9485  df-nr 9486  df-ltr 9489
This theorem is referenced by:  gt0srpr  9507  ltsosr  9523  0lt1sr  9524  ltasr  9529  mappsrpr  9537  ltpsrpr  9538  map2psrpr  9539
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