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Theorem ltasr 9473
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 9458 . 2  |-  dom  +R  =  ( R.  X.  R. )
2 ltrelsr 9441 . 2  |-  <R  C_  ( R.  X.  R. )
3 0nsr 9452 . 2  |-  -.  (/)  e.  R.
4 df-nr 9430 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 oveq1 6289 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. x ,  y >. ]  ~R  )  =  ( C  +R  [ <. x ,  y >. ]  ~R  ) )
6 oveq1 6289 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  =  ( C  +R  [ <. z ,  w >. ]  ~R  ) )
75, 6breq12d 4460 . . . . 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 318 . . . 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 4450 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
10 oveq2 6290 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( C  +R  [ <. x ,  y >. ]  ~R  )  =  ( C  +R  A ) )
1110breq1d 4457 . . . . 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 321 . . . 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 4451 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
14 oveq2 6290 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( C  +R  [ <. z ,  w >. ]  ~R  )  =  ( C  +R  B ) )
1514breq2d 4459 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  B
) ) )
1613, 15bibi12d 321 . . . 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 9392 . . . . . . 7  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  +P.  u
)  e.  P. )
18173ad2ant1 1017 . . . . . 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 9419 . . . . . . 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 9450 . . . . . . 7  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
21 ltsrpr 9450 . . . . . . . 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 3116 . . . . . . . . . 10  |-  v  e. 
_V
23 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
24 vex 3116 . . . . . . . . . 10  |-  u  e. 
_V
25 addcompr 9395 . . . . . . . . . 10  |-  ( y  +P.  z )  =  ( z  +P.  y
)
26 addasspr 9396 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  f )  =  ( y  +P.  (
z  +P.  f )
)
27 vex 3116 . . . . . . . . . 10  |-  w  e. 
_V
2822, 23, 24, 25, 26, 27caov4 6488 . . . . . . . . 9  |-  ( ( v  +P.  x )  +P.  ( u  +P.  w ) )  =  ( ( v  +P.  u )  +P.  (
x  +P.  w )
)
29 addcompr 9395 . . . . . . . . . 10  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  z )  +P.  (
u  +P.  y )
)
30 vex 3116 . . . . . . . . . . 11  |-  z  e. 
_V
31 addcompr 9395 . . . . . . . . . . 11  |-  ( x  +P.  w )  =  ( w  +P.  x
)
32 addasspr 9396 . . . . . . . . . . 11  |-  ( ( x  +P.  w )  +P.  f )  =  ( x  +P.  (
w  +P.  f )
)
33 vex 3116 . . . . . . . . . . 11  |-  y  e. 
_V
3422, 30, 24, 31, 32, 33caov42 6490 . . . . . . . . . 10  |-  ( ( v  +P.  z )  +P.  ( u  +P.  y ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3529, 34eqtri 2496 . . . . . . . . 9  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3628, 35breq12i 4456 . . . . . . . 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 249 . . . . . . 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 288 . . . . . 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 9448 . . . . . . 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 1016 . . . . . 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 9448 . . . . . . 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 1015 . . . . . 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 4460 . . . . 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 256 . . . 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 7400 . . 3  |-  ( ( C  e.  R.  /\  A  e.  R.  /\  B  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
47463coml 1203 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
481, 2, 3, 47ndmovord 6447 1  |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447  (class class class)co 6282   [cec 7306   P.cnp 9233    +P. cpp 9235    <P cltp 9237    ~R cer 9238   R.cnr 9239    +R cplr 9243    <R cltr 9245
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 6574  ax-inf2 8054
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-plp 9357  df-ltp 9359  df-enr 9429  df-nr 9430  df-plr 9431  df-ltr 9433
This theorem is referenced by:  addgt0sr  9477  sqgt0sr  9479  mappsrpr  9481  ltpsrpr  9482  map2psrpr  9483  supsrlem  9484  axpre-ltadd  9540
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