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Theorem ltasr 8901
Description: Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltasr  |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )

Proof of Theorem ltasr
Dummy variables  x  y  z  w  v  u  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmaddsr 8886 . 2  |-  dom  +R  =  ( R.  X.  R. )
2 ltrelsr 8872 . 2  |-  <R  C_  ( R.  X.  R. )
3 0nsr 8880 . 2  |-  -.  (/)  e.  R.
4 df-nr 8861 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 oveq1 6020 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. x ,  y >. ]  ~R  )  =  ( C  +R  [ <. x ,  y >. ]  ~R  ) )
6 oveq1 6020 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  =  ( C  +R  [ <. z ,  w >. ]  ~R  ) )
75, 6breq12d 4159 . . . . 5  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) ) )
87bibi2d 310 . . . 4  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  ) )  <-> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) ) ) )
9 breq1 4149 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
10 oveq2 6021 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( C  +R  [ <. x ,  y >. ]  ~R  )  =  ( C  +R  A ) )
1110breq1d 4156 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( C  +R  [
<. x ,  y >. ]  ~R  )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  [ <. z ,  w >. ]  ~R  ) ) )
129, 11bibi12d 313 . . . 4  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  [ <. x ,  y
>. ]  ~R  )  <R 
( C  +R  [ <. z ,  w >. ]  ~R  ) )  <->  ( A  <R  [ <. z ,  w >. ]  ~R  <->  ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )
) ) )
13 breq2 4150 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
14 oveq2 6021 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( C  +R  [ <. z ,  w >. ]  ~R  )  =  ( C  +R  B ) )
1514breq2d 4158 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  B
) ) )
1613, 15bibi12d 313 . . . 4  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( A  <R  [
<. z ,  w >. ]  ~R  <->  ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )
)  <->  ( A  <R  B  <-> 
( C  +R  A
)  <R  ( C  +R  B ) ) ) )
17 addclpr 8821 . . . . . . 7  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  +P.  u
)  e.  P. )
18173ad2ant1 978 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( v  +P.  u )  e.  P. )
19 ltapr 8848 . . . . . . 7  |-  ( ( v  +P.  u )  e.  P.  ->  (
( x  +P.  w
)  <P  ( y  +P.  z )  <->  ( (
v  +P.  u )  +P.  ( x  +P.  w
) )  <P  (
( v  +P.  u
)  +P.  ( y  +P.  z ) ) ) )
20 ltsrpr 8878 . . . . . . 7  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
21 ltsrpr 8878 . . . . . . . 8  |-  ( [
<. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  <->  ( ( v  +P.  x
)  +P.  ( u  +P.  w ) )  <P 
( ( u  +P.  y )  +P.  (
v  +P.  z )
) )
22 vex 2895 . . . . . . . . . 10  |-  v  e. 
_V
23 vex 2895 . . . . . . . . . 10  |-  x  e. 
_V
24 vex 2895 . . . . . . . . . 10  |-  u  e. 
_V
25 addcompr 8824 . . . . . . . . . 10  |-  ( y  +P.  z )  =  ( z  +P.  y
)
26 addasspr 8825 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  f )  =  ( y  +P.  (
z  +P.  f )
)
27 vex 2895 . . . . . . . . . 10  |-  w  e. 
_V
2822, 23, 24, 25, 26, 27caov4 6210 . . . . . . . . 9  |-  ( ( v  +P.  x )  +P.  ( u  +P.  w ) )  =  ( ( v  +P.  u )  +P.  (
x  +P.  w )
)
29 addcompr 8824 . . . . . . . . . 10  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  z )  +P.  (
u  +P.  y )
)
30 vex 2895 . . . . . . . . . . 11  |-  z  e. 
_V
31 addcompr 8824 . . . . . . . . . . 11  |-  ( x  +P.  w )  =  ( w  +P.  x
)
32 addasspr 8825 . . . . . . . . . . 11  |-  ( ( x  +P.  w )  +P.  f )  =  ( x  +P.  (
w  +P.  f )
)
33 vex 2895 . . . . . . . . . . 11  |-  y  e. 
_V
3422, 30, 24, 31, 32, 33caov42 6212 . . . . . . . . . 10  |-  ( ( v  +P.  z )  +P.  ( u  +P.  y ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3529, 34eqtri 2400 . . . . . . . . 9  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3628, 35breq12i 4155 . . . . . . . 8  |-  ( ( ( v  +P.  x
)  +P.  ( u  +P.  w ) )  <P 
( ( u  +P.  y )  +P.  (
v  +P.  z )
)  <->  ( ( v  +P.  u )  +P.  ( x  +P.  w
) )  <P  (
( v  +P.  u
)  +P.  ( y  +P.  z ) ) )
3721, 36bitri 241 . . . . . . 7  |-  ( [
<. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  <->  ( ( v  +P.  u
)  +P.  ( x  +P.  w ) )  <P 
( ( v  +P.  u )  +P.  (
y  +P.  z )
) )
3819, 20, 373bitr4g 280 . . . . . 6  |-  ( ( v  +P.  u )  e.  P.  ->  ( [ <. x ,  y
>. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  [
<. ( v  +P.  x
) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  ) )
3918, 38syl 16 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. ( v  +P.  z ) ,  ( u  +P.  w )
>. ]  ~R  ) )
40 addsrpr 8876 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  =  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  )
41403adant3 977 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  =  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  )
42 addsrpr 8876 . . . . . . 7  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  )
43423adant2 976 . . . . . 6  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  =  [ <. (
v  +P.  z ) ,  ( u  +P.  w ) >. ]  ~R  )
4441, 43breq12d 4159 . . . . 5  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  )  <->  [ <. (
v  +P.  x ) ,  ( u  +P.  y ) >. ]  ~R  <R  [ <. ( v  +P.  z ) ,  ( u  +P.  w )
>. ]  ~R  ) )
4539, 44bitr4d 248 . . . 4  |-  ( ( ( v  e.  P.  /\  u  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( [ <. v ,  u >. ]  ~R  +R  [ <. x ,  y >. ]  ~R  )  <R  ( [ <. v ,  u >. ]  ~R  +R  [ <. z ,  w >. ]  ~R  ) ) )
464, 8, 12, 16, 453ecoptocl 6925 . . 3  |-  ( ( C  e.  R.  /\  A  e.  R.  /\  B  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
47463coml 1160 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
481, 2, 3, 47ndmovord 6169 1  |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146  (class class class)co 6013   [cec 6832   P.cnp 8660    +P. cpp 8662    <P cltp 8664    ~R cer 8667   R.cnr 8668    +R cplr 8672    <R cltr 8674
This theorem is referenced by:  addgt0sr  8905  sqgt0sr  8907  mappsrpr  8909  ltpsrpr  8910  map2psrpr  8911  supsrlem  8912  axpre-ltadd  8968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-omul 6658  df-er 6834  df-ec 6836  df-qs 6840  df-ni 8675  df-pli 8676  df-mi 8677  df-lti 8678  df-plpq 8711  df-mpq 8712  df-ltpq 8713  df-enq 8714  df-nq 8715  df-erq 8716  df-plq 8717  df-mq 8718  df-1nq 8719  df-rq 8720  df-ltnq 8721  df-np 8784  df-plp 8786  df-ltp 8788  df-plpr 8858  df-enr 8860  df-nr 8861  df-plr 8862  df-ltr 8864
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