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Theorem ltasr 9509
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 9494 . 2  |-  dom  +R  =  ( R.  X.  R. )
2 ltrelsr 9477 . 2  |-  <R  C_  ( R.  X.  R. )
3 0nsr 9488 . 2  |-  -.  (/)  e.  R.
4 df-nr 9466 . . . 4  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5 oveq1 6287 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. x ,  y >. ]  ~R  )  =  ( C  +R  [ <. x ,  y >. ]  ~R  ) )
6 oveq1 6287 . . . . . 6  |-  ( [
<. v ,  u >. ]  ~R  =  C  -> 
( [ <. v ,  u >. ]  ~R  +R  [
<. z ,  w >. ]  ~R  )  =  ( C  +R  [ <. z ,  w >. ]  ~R  ) )
75, 6breq12d 4410 . . . . 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 4400 . . . . 5  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  A  <R  [
<. z ,  w >. ]  ~R  ) )
10 oveq2 6288 . . . . . 6  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( C  +R  [ <. x ,  y >. ]  ~R  )  =  ( C  +R  A ) )
1110breq1d 4407 . . . . 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 4401 . . . . 5  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( A  <R  [ <. z ,  w >. ]  ~R  <->  A 
<R  B ) )
14 oveq2 6288 . . . . . 6  |-  ( [
<. z ,  w >. ]  ~R  =  B  -> 
( C  +R  [ <. z ,  w >. ]  ~R  )  =  ( C  +R  B ) )
1514breq2d 4409 . . . . 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 9428 . . . . . . 7  |-  ( ( v  e.  P.  /\  u  e.  P. )  ->  ( v  +P.  u
)  e.  P. )
18173ad2ant1 1020 . . . . . 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 9455 . . . . . . 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 9486 . . . . . . 7  |-  ( [
<. x ,  y >. ]  ~R  <R  [ <. z ,  w >. ]  ~R  <->  ( x  +P.  w )  <P  (
y  +P.  z )
)
21 ltsrpr 9486 . . . . . . . 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 3064 . . . . . . . . . 10  |-  v  e. 
_V
23 vex 3064 . . . . . . . . . 10  |-  x  e. 
_V
24 vex 3064 . . . . . . . . . 10  |-  u  e. 
_V
25 addcompr 9431 . . . . . . . . . 10  |-  ( y  +P.  z )  =  ( z  +P.  y
)
26 addasspr 9432 . . . . . . . . . 10  |-  ( ( y  +P.  z )  +P.  f )  =  ( y  +P.  (
z  +P.  f )
)
27 vex 3064 . . . . . . . . . 10  |-  w  e. 
_V
2822, 23, 24, 25, 26, 27caov4 6489 . . . . . . . . 9  |-  ( ( v  +P.  x )  +P.  ( u  +P.  w ) )  =  ( ( v  +P.  u )  +P.  (
x  +P.  w )
)
29 addcompr 9431 . . . . . . . . . 10  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  z )  +P.  (
u  +P.  y )
)
30 vex 3064 . . . . . . . . . . 11  |-  z  e. 
_V
31 addcompr 9431 . . . . . . . . . . 11  |-  ( x  +P.  w )  =  ( w  +P.  x
)
32 addasspr 9432 . . . . . . . . . . 11  |-  ( ( x  +P.  w )  +P.  f )  =  ( x  +P.  (
w  +P.  f )
)
33 vex 3064 . . . . . . . . . . 11  |-  y  e. 
_V
3422, 30, 24, 31, 32, 33caov42 6491 . . . . . . . . . 10  |-  ( ( v  +P.  z )  +P.  ( u  +P.  y ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3529, 34eqtri 2433 . . . . . . . . 9  |-  ( ( u  +P.  y )  +P.  ( v  +P.  z ) )  =  ( ( v  +P.  u )  +P.  (
y  +P.  z )
)
3628, 35breq12i 4406 . . . . . . . 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 251 . . . . . . 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 290 . . . . . 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 17 . . . . 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 9484 . . . . . . 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 1019 . . . . . 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 9484 . . . . . . 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 1018 . . . . . 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 4410 . . . . 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 258 . . . 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 7442 . . 3  |-  ( ( C  e.  R.  /\  A  e.  R.  /\  B  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
47463coml 1206 . 2  |-  ( ( A  e.  R.  /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
481, 2, 3, 47ndmovord 6448 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   <.cop 3980   class class class wbr 4397  (class class class)co 6280   [cec 7348   P.cnp 9269    +P. cpp 9271    <P cltp 9273    ~R cer 9274   R.cnr 9275    +R cplr 9279    <R cltr 9281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-omul 7174  df-er 7350  df-ec 7352  df-qs 7356  df-ni 9282  df-pli 9283  df-mi 9284  df-lti 9285  df-plpq 9318  df-mpq 9319  df-ltpq 9320  df-enq 9321  df-nq 9322  df-erq 9323  df-plq 9324  df-mq 9325  df-1nq 9326  df-rq 9327  df-ltnq 9328  df-np 9391  df-plp 9393  df-ltp 9395  df-enr 9465  df-nr 9466  df-plr 9467  df-ltr 9469
This theorem is referenced by:  addgt0sr  9513  sqgt0sr  9515  mappsrpr  9517  ltpsrpr  9518  map2psrpr  9519  supsrlem  9520  axpre-ltadd  9576
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