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Theorem ltsrpr 9452
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 9442 . 2  |-  ~R  e.  _V
2 enrer 9440 . . 3  |-  ~R  Er  ( P.  X.  P. )
3 erdm 7319 . . 3  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
42, 3ax-mp 5 . 2  |-  dom  ~R  =  ( P.  X.  P. )
5 df-nr 9432 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6 ltrelsr 9443 . 2  |-  <R  C_  ( R.  X.  R. )
7 ltrelpr 9374 . 2  |-  <P  C_  ( P.  X.  P. )
8 0npr 9368 . 2  |-  -.  (/)  e.  P.
9 dmplp 9388 . 2  |-  dom  +P.  =  ( P.  X.  P. )
10 df-ltr 9435 . . 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 9394 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
1211ad2ant2lr 747 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
13 addclpr 9394 . . . . . . 7  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
1413ad2ant2lr 747 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
1512, 14anim12ci 567 . . . . 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 824 . . . 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 9441 . . . . . 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 9441 . . . . . . 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 2450 . . . . . . 7  |-  ( ( v  +P.  D )  =  ( u  +P.  C )  <->  ( u  +P.  C )  =  ( v  +P.  D ) )
2018, 19syl6bb 261 . . . . . 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 871 . . . . 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 6286 . . . . . 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 9397 . . . . . . . . . 10  |-  ( u  +P.  B )  =  ( B  +P.  u
)
2423oveq1i 6287 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P.  C
)
25 addasspr 9398 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( u  +P.  ( B  +P.  C ) )
26 addasspr 9398 . . . . . . . . 9  |-  ( ( B  +P.  u )  +P.  C )  =  ( B  +P.  (
u  +P.  C )
)
2724, 25, 263eqtr3i 2478 . . . . . . . 8  |-  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) )
2827oveq2i 6288 . . . . . . 7  |-  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
29 addasspr 9398 . . . . . . 7  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )
30 addasspr 9398 . . . . . . 7  |-  ( ( z  +P.  B )  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
3128, 29, 303eqtr4i 2480 . . . . . 6  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) )
32 addcompr 9397 . . . . . . . . . 10  |-  ( v  +P.  A )  =  ( A  +P.  v
)
3332oveq1i 6287 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( ( A  +P.  v )  +P.  D
)
34 addasspr 9398 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( v  +P.  ( A  +P.  D ) )
35 addasspr 9398 . . . . . . . . 9  |-  ( ( A  +P.  v )  +P.  D )  =  ( A  +P.  (
v  +P.  D )
)
3633, 34, 353eqtr3i 2478 . . . . . . . 8  |-  ( v  +P.  ( A  +P.  D ) )  =  ( A  +P.  ( v  +P.  D ) )
3736oveq2i 6288 . . . . . . 7  |-  ( w  +P.  ( v  +P.  ( A  +P.  D
) ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
38 addasspr 9398 . . . . . . 7  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )
39 addasspr 9398 . . . . . . 7  |-  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
4037, 38, 393eqtr4i 2480 . . . . . 6  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) )
4122, 31, 403eqtr4g 2507 . . . . 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 228 . . . 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 6305 . . . . 5  |-  ( z  +P.  u )  e. 
_V
44 ovex 6305 . . . . 5  |-  ( B  +P.  C )  e. 
_V
45 ltapr 9421 . . . . 5  |-  ( f  e.  P.  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
46 ovex 6305 . . . . 5  |-  ( w  +P.  v )  e. 
_V
47 addcompr 9397 . . . . 5  |-  ( x  +P.  y )  =  ( y  +P.  x
)
48 ovex 6305 . . . . 5  |-  ( A  +P.  D )  e. 
_V
4943, 44, 45, 46, 47, 48caovord3 6469 . . . 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 545 . . 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 7402 . 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 7403 1  |-  ( [
<. A ,  B >. ]  ~R  <R  [ <. C ,  D >. ]  ~R  <->  ( A  +P.  D )  <P  ( B  +P.  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   <.cop 4016   class class class wbr 4433    X. cxp 4983   dom cdm 4985  (class class class)co 6277    Er wer 7306   [cec 7307   P.cnp 9235    +P. cpp 9237    <P cltp 9239    ~R cer 9240   R.cnr 9241    <R cltr 9247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-omul 7133  df-er 7309  df-ec 7311  df-qs 7315  df-ni 9248  df-pli 9249  df-mi 9250  df-lti 9251  df-plpq 9284  df-mpq 9285  df-ltpq 9286  df-enq 9287  df-nq 9288  df-erq 9289  df-plq 9290  df-mq 9291  df-1nq 9292  df-rq 9293  df-ltnq 9294  df-np 9357  df-plp 9359  df-ltp 9361  df-enr 9431  df-nr 9432  df-ltr 9435
This theorem is referenced by:  gt0srpr  9453  ltsosr  9469  0lt1sr  9470  ltasr  9475  mappsrpr  9483  ltpsrpr  9484  map2psrpr  9485
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