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Theorem ltsrpr 9383
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 9373 . 2  |-  ~R  e.  _V
2 enrer 9371 . . 3  |-  ~R  Er  ( P.  X.  P. )
3 erdm 7257 . . 3  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
42, 3ax-mp 5 . 2  |-  dom  ~R  =  ( P.  X.  P. )
5 df-nr 9363 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6 ltrelsr 9374 . 2  |-  <R  C_  ( R.  X.  R. )
7 ltrelpr 9305 . 2  |-  <P  C_  ( P.  X.  P. )
8 0npr 9299 . 2  |-  -.  (/)  e.  P.
9 dmplp 9319 . 2  |-  dom  +P.  =  ( P.  X.  P. )
10 df-ltr 9366 . . 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 9325 . . . . . . 7  |-  ( ( w  e.  P.  /\  v  e.  P. )  ->  ( w  +P.  v
)  e.  P. )
1211ad2ant2lr 745 . . . . . 6  |-  ( ( ( z  e.  P.  /\  w  e.  P. )  /\  ( v  e.  P.  /\  u  e.  P. )
)  ->  ( w  +P.  v )  e.  P. )
13 addclpr 9325 . . . . . . 7  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B  +P.  C
)  e.  P. )
1413ad2ant2lr 745 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  +P.  C )  e.  P. )
1512, 14anim12ci 565 . . . . 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 9372 . . . . . 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 9372 . . . . . . 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 2401 . . . . . . 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 6223 . . . . . 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 9328 . . . . . . . . . 10  |-  ( u  +P.  B )  =  ( B  +P.  u
)
2423oveq1i 6224 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( ( B  +P.  u )  +P.  C
)
25 addasspr 9329 . . . . . . . . 9  |-  ( ( u  +P.  B )  +P.  C )  =  ( u  +P.  ( B  +P.  C ) )
26 addasspr 9329 . . . . . . . . 9  |-  ( ( B  +P.  u )  +P.  C )  =  ( B  +P.  (
u  +P.  C )
)
2724, 25, 263eqtr3i 2429 . . . . . . . 8  |-  ( u  +P.  ( B  +P.  C ) )  =  ( B  +P.  ( u  +P.  C ) )
2827oveq2i 6225 . . . . . . 7  |-  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
29 addasspr 9329 . . . . . . 7  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( z  +P.  ( u  +P.  ( B  +P.  C ) ) )
30 addasspr 9329 . . . . . . 7  |-  ( ( z  +P.  B )  +P.  ( u  +P.  C ) )  =  ( z  +P.  ( B  +P.  ( u  +P.  C ) ) )
3128, 29, 303eqtr4i 2431 . . . . . 6  |-  ( ( z  +P.  u )  +P.  ( B  +P.  C ) )  =  ( ( z  +P.  B
)  +P.  ( u  +P.  C ) )
32 addcompr 9328 . . . . . . . . . 10  |-  ( v  +P.  A )  =  ( A  +P.  v
)
3332oveq1i 6224 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( ( A  +P.  v )  +P.  D
)
34 addasspr 9329 . . . . . . . . 9  |-  ( ( v  +P.  A )  +P.  D )  =  ( v  +P.  ( A  +P.  D ) )
35 addasspr 9329 . . . . . . . . 9  |-  ( ( A  +P.  v )  +P.  D )  =  ( A  +P.  (
v  +P.  D )
)
3633, 34, 353eqtr3i 2429 . . . . . . . 8  |-  ( v  +P.  ( A  +P.  D ) )  =  ( A  +P.  ( v  +P.  D ) )
3736oveq2i 6225 . . . . . . 7  |-  ( w  +P.  ( v  +P.  ( A  +P.  D
) ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
38 addasspr 9329 . . . . . . 7  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( w  +P.  ( v  +P.  ( A  +P.  D ) ) )
39 addasspr 9329 . . . . . . 7  |-  ( ( w  +P.  A )  +P.  ( v  +P. 
D ) )  =  ( w  +P.  ( A  +P.  ( v  +P. 
D ) ) )
4037, 38, 393eqtr4i 2431 . . . . . 6  |-  ( ( w  +P.  v )  +P.  ( A  +P.  D ) )  =  ( ( w  +P.  A
)  +P.  ( v  +P.  D ) )
4122, 31, 403eqtr4g 2458 . . . . 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 6242 . . . . 5  |-  ( z  +P.  u )  e. 
_V
44 ovex 6242 . . . . 5  |-  ( B  +P.  C )  e. 
_V
45 ltapr 9352 . . . . 5  |-  ( f  e.  P.  ->  (
x  <P  y  <->  ( f  +P.  x )  <P  (
f  +P.  y )
) )
46 ovex 6242 . . . . 5  |-  ( w  +P.  v )  e. 
_V
47 addcompr 9328 . . . . 5  |-  ( x  +P.  y )  =  ( y  +P.  x
)
48 ovex 6242 . . . . 5  |-  ( A  +P.  D )  e. 
_V
4943, 44, 45, 46, 47, 48caovord3 6405 . . . 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 543 . . 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 7340 . 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 7341 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 367    = wceq 1399    e. wcel 1836   <.cop 3963   class class class wbr 4380    X. cxp 4924   dom cdm 4926  (class class class)co 6214    Er wer 7244   [cec 7245   P.cnp 9166    +P. cpp 9168    <P cltp 9170    ~R cer 9171   R.cnr 9172    <R cltr 9178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-inf2 7990
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-uni 4177  df-int 4213  df-iun 4258  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-om 6618  df-1st 6717  df-2nd 6718  df-recs 6978  df-rdg 7012  df-1o 7066  df-oadd 7070  df-omul 7071  df-er 7247  df-ec 7249  df-qs 7253  df-ni 9179  df-pli 9180  df-mi 9181  df-lti 9182  df-plpq 9215  df-mpq 9216  df-ltpq 9217  df-enq 9218  df-nq 9219  df-erq 9220  df-plq 9221  df-mq 9222  df-1nq 9223  df-rq 9224  df-ltnq 9225  df-np 9288  df-plp 9290  df-ltp 9292  df-enr 9362  df-nr 9363  df-ltr 9366
This theorem is referenced by:  gt0srpr  9384  ltsosr  9400  0lt1sr  9401  ltasr  9406  mappsrpr  9414  ltpsrpr  9415  map2psrpr  9416
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