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Theorem 00sr 7218
Description: A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
Assertion
Ref Expression
00sr  |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )

Proof of Theorem 00sr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 7176 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5598 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R  0R )  =  ( A  .R  0R ) )
32eqeq1d 2091 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R  <->  ( A  .R  0R )  =  0R ) )
4 1pr 7016 . . . . 5  |-  1P  e.  P.
5 mulsrpr 7195 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( 1P  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  .R  [ <. 1P ,  1P >. ]  ~R  )  =  [ <. (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )
>. ]  ~R  )
64, 4, 5mpanr12 430 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )  =  [ <. ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) >. ]  ~R  )
7 mulclpr 7034 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
84, 7mpan2 416 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
9 mulclpr 7034 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
104, 9mpan2 416 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
11 addclpr 6999 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  e. 
P. )
128, 10, 11syl2an 283 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )
1312, 12anim12i 331 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  e. 
P.  /\  ( (
x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. ) )
14 eqid 2083 . . . . . . . 8  |-  ( ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  +P. 
1P )  =  ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  +P.  1P )
15 enreceq 7185 . . . . . . . 8  |-  ( ( ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P.  /\  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )  /\  ( 1P  e.  P.  /\  1P  e.  P. )
)  ->  ( [ <. ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) ) >. ]  ~R  =  [ <. 1P ,  1P >. ]  ~R  <->  ( (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  +P. 
1P )  =  ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  +P.  1P ) ) )
1614, 15mpbiri 166 . . . . . . 7  |-  ( ( ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P.  /\  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )  /\  ( 1P  e.  P.  /\  1P  e.  P. )
)  ->  [ <. (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )
>. ]  ~R  =  [ <. 1P ,  1P >. ]  ~R  )
1713, 16sylan 277 . . . . . 6  |-  ( ( ( ( x  e. 
P.  /\  y  e.  P. )  /\  (
x  e.  P.  /\  y  e.  P. )
)  /\  ( 1P  e.  P.  /\  1P  e.  P. ) )  ->  [ <. ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )
>. ]  ~R  =  [ <. 1P ,  1P >. ]  ~R  )
184, 4, 17mpanr12 430 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  [ <. (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )
>. ]  ~R  =  [ <. 1P ,  1P >. ]  ~R  )
1918anidms 389 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  [ <. ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) ) >. ]  ~R  =  [ <. 1P ,  1P >. ]  ~R  )
206, 19eqtrd 2115 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R  )
21 df-0r 7180 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
2221oveq2i 5602 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  0R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )
2320, 22, 213eqtr4g 2140 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R )
241, 3, 23ecoptocl 6309 1  |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   <.cop 3425  (class class class)co 5591   [cec 6220   P.cnp 6753   1Pc1p 6754    +P. cpp 6755    .P. cmp 6756    ~R cer 6758   R.cnr 6759   0Rc0r 6760    .R cmr 6764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-i1p 6929  df-iplp 6930  df-imp 6931  df-enr 7175  df-nr 7176  df-mr 7178  df-0r 7180
This theorem is referenced by:  pn0sr  7220  mulresr  7278  axi2m1  7313  axcnre  7319
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