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Theorem ltsrpr 9265
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 9258 . 2  |-  ~R  e.  _V
2 enrer 9256 . . 3  |-  ~R  Er  ( P.  X.  P. )
3 erdm 7132 . . 3  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
42, 3ax-mp 5 . 2  |-  dom  ~R  =  ( P.  X.  P. )
5 df-nr 9248 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6 ltrelsr 9259 . 2  |-  <R  C_  ( R.  X.  R. )
7 ltrelpr 9188 . 2  |-  <P  C_  ( P.  X.  P. )
8 0npr 9182 . 2  |-  -.  (/)  e.  P.
9 dmplp 9202 . 2  |-  dom  +P.  =  ( P.  X.  P. )
10 df-ltr 9251 . . 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 9208 . . . . . . 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 9208 . . . . . . 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 822 . . . 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 9257 . . . . . 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 9257 . . . . . . 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 2445 . . . . . . 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 868 . . . . 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 6121 . . . . . 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 9211 . . . . . . . . . 10  |-  ( u  +P.  B )  =  ( B  +P.  u
)
2423oveq1i 6122 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P.  C
)
25 addasspr 9212 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( u  +P.  ( B  +P.  C ) )
26 addasspr 9212 . . . . . . . . 9  |-  ( ( B  +P.  u )  +P.  C )  =  ( B  +P.  (
u  +P.  C )
)
2724, 25, 263eqtr3i 2471 . . . . . . . 8  |-  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) )
2827oveq2i 6123 . . . . . . 7  |-  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
29 addasspr 9212 . . . . . . 7  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )
30 addasspr 9212 . . . . . . 7  |-  ( ( z  +P.  B )  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
3128, 29, 303eqtr4i 2473 . . . . . 6  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) )
32 addcompr 9211 . . . . . . . . . 10  |-  ( v  +P.  A )  =  ( A  +P.  v
)
3332oveq1i 6122 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( ( A  +P.  v )  +P.  D
)
34 addasspr 9212 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( v  +P.  ( A  +P.  D ) )
35 addasspr 9212 . . . . . . . . 9  |-  ( ( A  +P.  v )  +P.  D )  =  ( A  +P.  (
v  +P.  D )
)
3633, 34, 353eqtr3i 2471 . . . . . . . 8  |-  ( v  +P.  ( A  +P.  D ) )  =  ( A  +P.  ( v  +P.  D ) )
3736oveq2i 6123 . . . . . . 7  |-  ( w  +P.  ( v  +P.  ( A  +P.  D
) ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
38 addasspr 9212 . . . . . . 7  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )
39 addasspr 9212 . . . . . . 7  |-  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
4037, 38, 393eqtr4i 2473 . . . . . 6  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) )
4122, 31, 403eqtr4g 2500 . . . . 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 6137 . . . . 5  |-  ( z  +P.  u )  e. 
_V
44 ovex 6137 . . . . 5  |-  ( B  +P.  C )  e. 
_V
45 ltapr 9235 . . . . 5  |-  ( f  e.  P.  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
46 ovex 6137 . . . . 5  |-  ( w  +P.  v )  e. 
_V
47 addcompr 9211 . . . . 5  |-  ( x  +P.  y )  =  ( y  +P.  x
)
48 ovex 6137 . . . . 5  |-  ( A  +P.  D )  e. 
_V
4943, 44, 45, 46, 47, 48caovord3 6297 . . . 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 7214 . 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 7215 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 1369    e. wcel 1756   <.cop 3904   class class class wbr 4313    X. cxp 4859   dom cdm 4861  (class class class)co 6112    Er wer 7119   [cec 7120   P.cnp 9047    +P. cpp 9049    <P cltp 9051    ~R cer 9054   R.cnr 9055    <R cltr 9061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-ltp 9175  df-enr 9247  df-nr 9248  df-ltr 9251
This theorem is referenced by:  gt0srpr  9266  ltsosr  9282  0lt1sr  9283  ltasr  9288  mappsrpr  9296  ltpsrpr  9297  map2psrpr  9298
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