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Theorem ltsrpr 9455
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 9445 . 2  |-  ~R  e.  _V
2 enrer 9443 . . 3  |-  ~R  Er  ( P.  X.  P. )
3 erdm 7322 . . 3  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
42, 3ax-mp 5 . 2  |-  dom  ~R  =  ( P.  X.  P. )
5 df-nr 9435 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6 ltrelsr 9446 . 2  |-  <R  C_  ( R.  X.  R. )
7 ltrelpr 9377 . 2  |-  <P  C_  ( P.  X.  P. )
8 0npr 9371 . 2  |-  -.  (/)  e.  P.
9 dmplp 9391 . 2  |-  dom  +P.  =  ( P.  X.  P. )
10 df-ltr 9438 . . 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 9397 . . . . . . 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 9397 . . . . . . 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 9444 . . . . . 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 9444 . . . . . . 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 2476 . . . . . . 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 6294 . . . . . 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 9400 . . . . . . . . . 10  |-  ( u  +P.  B )  =  ( B  +P.  u
)
2423oveq1i 6295 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P.  C
)
25 addasspr 9401 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( u  +P.  ( B  +P.  C ) )
26 addasspr 9401 . . . . . . . . 9  |-  ( ( B  +P.  u )  +P.  C )  =  ( B  +P.  (
u  +P.  C )
)
2724, 25, 263eqtr3i 2504 . . . . . . . 8  |-  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) )
2827oveq2i 6296 . . . . . . 7  |-  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
29 addasspr 9401 . . . . . . 7  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )
30 addasspr 9401 . . . . . . 7  |-  ( ( z  +P.  B )  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
3128, 29, 303eqtr4i 2506 . . . . . 6  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) )
32 addcompr 9400 . . . . . . . . . 10  |-  ( v  +P.  A )  =  ( A  +P.  v
)
3332oveq1i 6295 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( ( A  +P.  v )  +P.  D
)
34 addasspr 9401 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( v  +P.  ( A  +P.  D ) )
35 addasspr 9401 . . . . . . . . 9  |-  ( ( A  +P.  v )  +P.  D )  =  ( A  +P.  (
v  +P.  D )
)
3633, 34, 353eqtr3i 2504 . . . . . . . 8  |-  ( v  +P.  ( A  +P.  D ) )  =  ( A  +P.  ( v  +P.  D ) )
3736oveq2i 6296 . . . . . . 7  |-  ( w  +P.  ( v  +P.  ( A  +P.  D
) ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
38 addasspr 9401 . . . . . . 7  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )
39 addasspr 9401 . . . . . . 7  |-  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
4037, 38, 393eqtr4i 2506 . . . . . 6  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) )
4122, 31, 403eqtr4g 2533 . . . . 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 6310 . . . . 5  |-  ( z  +P.  u )  e. 
_V
44 ovex 6310 . . . . 5  |-  ( B  +P.  C )  e. 
_V
45 ltapr 9424 . . . . 5  |-  ( f  e.  P.  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
46 ovex 6310 . . . . 5  |-  ( w  +P.  v )  e. 
_V
47 addcompr 9400 . . . . 5  |-  ( x  +P.  y )  =  ( y  +P.  x
)
48 ovex 6310 . . . . 5  |-  ( A  +P.  D )  e. 
_V
4943, 44, 45, 46, 47, 48caovord3 6473 . . . 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 7405 . 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 7406 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 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    X. cxp 4997   dom cdm 4999  (class class class)co 6285    Er wer 7309   [cec 7310   P.cnp 9238    +P. cpp 9240    <P cltp 9242    ~R cer 9243   R.cnr 9244    <R cltr 9250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-omul 7136  df-er 7312  df-ec 7314  df-qs 7318  df-ni 9251  df-pli 9252  df-mi 9253  df-lti 9254  df-plpq 9287  df-mpq 9288  df-ltpq 9289  df-enq 9290  df-nq 9291  df-erq 9292  df-plq 9293  df-mq 9294  df-1nq 9295  df-rq 9296  df-ltnq 9297  df-np 9360  df-plp 9362  df-ltp 9364  df-enr 9434  df-nr 9435  df-ltr 9438
This theorem is referenced by:  gt0srpr  9456  ltsosr  9472  0lt1sr  9473  ltasr  9478  mappsrpr  9486  ltpsrpr  9487  map2psrpr  9488
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