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Theorem ltasr 8738
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 8723 . 2  |-  dom  +R  =  ( R.  X.  R. )
2 ltrelsr 8709 . 2  |-  <R  C_  ( R.  X.  R. )
3 0nsr 8717 . 2  |-  -.  (/)  e.  R.
4 df-nr 8698 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 oveq1 5881 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. x ,  y >. ]  ~R  )  =  ( C  +R  [ <. x ,  y >. ]  ~R  ) )
6 oveq1 5881 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  =  ( C  +R  [ <. z ,  w >. ]  ~R  ) )
75, 6breq12d 4052 . . . . 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 309 . . . 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 4042 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
10 oveq2 5882 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( C  +R  [ <. x ,  y >. ]  ~R  )  =  ( C  +R  A ) )
1110breq1d 4049 . . . . 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 312 . . . 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 4043 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
14 oveq2 5882 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( C  +R  [ <. z ,  w >. ]  ~R  )  =  ( C  +R  B ) )
1514breq2d 4051 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( ( C  +R  A )  <R  ( C  +R  [ <. z ,  w >. ]  ~R  )  <->  ( C  +R  A ) 
<R  ( C  +R  B
) ) )
1613, 15bibi12d 312 . . . 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 8658 . . . . . . 7  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  +P.  u
)  e.  P. )
18173ad2ant1 976 . . . . . 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 8685 . . . . . . 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 8715 . . . . . . 7  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
21 ltsrpr 8715 . . . . . . . 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 2804 . . . . . . . . . 10  |-  v  e. 
_V
23 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
24 vex 2804 . . . . . . . . . 10  |-  u  e. 
_V
25 addcompr 8661 . . . . . . . . . 10  |-  ( y  +P.  z )  =  ( z  +P.  y
)
26 addasspr 8662 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  f )  =  ( y  +P.  (
z  +P.  f )
)
27 vex 2804 . . . . . . . . . 10  |-  w  e. 
_V
2822, 23, 24, 25, 26, 27caov4 6067 . . . . . . . . 9  |-  ( ( v  +P.  x )  +P.  ( u  +P.  w ) )  =  ( ( v  +P.  u )  +P.  (
x  +P.  w )
)
29 addcompr 8661 . . . . . . . . . 10  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  z )  +P.  (
u  +P.  y )
)
30 vex 2804 . . . . . . . . . . 11  |-  z  e. 
_V
31 addcompr 8661 . . . . . . . . . . 11  |-  ( x  +P.  w )  =  ( w  +P.  x
)
32 addasspr 8662 . . . . . . . . . . 11  |-  ( ( x  +P.  w )  +P.  f )  =  ( x  +P.  (
w  +P.  f )
)
33 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
3422, 30, 24, 31, 32, 33caov42 6069 . . . . . . . . . 10  |-  ( ( v  +P.  z )  +P.  ( u  +P.  y ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3529, 34eqtri 2316 . . . . . . . . 9  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3628, 35breq12i 4048 . . . . . . . 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 240 . . . . . . 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 279 . . . . . 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 15 . . . . 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 8713 . . . . . . 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 975 . . . . . 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 8713 . . . . . . 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 974 . . . . . 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 4052 . . . . 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 247 . . . 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 6766 . . 3  |-  ( ( C  e.  R.  /\  A  e.  R.  /\  B  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
47463coml 1158 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
481, 2, 3, 47ndmovord 6026 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039  (class class class)co 5874   [cec 6674   P.cnp 8497    +P. cpp 8499    <P cltp 8501    ~R cer 8504   R.cnr 8505    +R cplr 8509    <R cltr 8511
This theorem is referenced by:  addgt0sr  8742  sqgt0sr  8744  mappsrpr  8746  ltpsrpr  8747  map2psrpr  8748  supsrlem  8749  axpre-ltadd  8805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-plp 8623  df-ltp 8625  df-plpr 8695  df-enr 8697  df-nr 8698  df-plr 8699  df-ltr 8701
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