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Theorem 00sr 9424
Description: A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
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 9382 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6239 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R  0R )  =  ( A  .R  0R ) )
32eqeq1d 2402 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R  <->  ( A  .R  0R )  =  0R ) )
4 1pr 9341 . . . . 5  |-  1P  e.  P.
5 mulsrpr 9401 . . . . 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 683 . . . 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 9346 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
84, 7mpan2 669 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
9 mulclpr 9346 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
104, 9mpan2 669 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
11 addclpr 9344 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  e. 
P. )
128, 10, 11syl2an 475 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )
1312, 12anim12i 564 . . . . . . 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 2400 . . . . . . . 8  |-  ( ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  +P. 
1P )  =  ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  +P.  1P )
15 enreceq 9391 . . . . . . . 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 233 . . . . . . 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 469 . . . . . 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 683 . . . . 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 643 . . . 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 2441 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R  )
21 df-0r 9386 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
2221oveq2i 6243 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  0R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )
2320, 22, 213eqtr4g 2466 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R )
241, 3, 23ecoptocl 7356 1  |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   <.cop 3975  (class class class)co 6232   [cec 7264   P.cnp 9185   1Pc1p 9186    +P. cpp 9187    .P. cmp 9188    ~R cer 9190   R.cnr 9191   0Rc0r 9192    .R cmr 9196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-omul 7090  df-er 7266  df-ec 7268  df-qs 7272  df-ni 9198  df-pli 9199  df-mi 9200  df-lti 9201  df-plpq 9234  df-mpq 9235  df-ltpq 9236  df-enq 9237  df-nq 9238  df-erq 9239  df-plq 9240  df-mq 9241  df-1nq 9242  df-rq 9243  df-ltnq 9244  df-np 9307  df-1p 9308  df-plp 9309  df-mp 9310  df-ltp 9311  df-enr 9381  df-nr 9382  df-mr 9384  df-0r 9386
This theorem is referenced by:  pn0sr  9426  mulresr  9464  axi2m1  9484  axcnre  9489
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