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Theorem 00sr 9380
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 9341 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6210 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R  0R )  =  ( A  .R  0R ) )
32eqeq1d 2456 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R  <->  ( A  .R  0R )  =  0R ) )
4 1pr 9298 . . . . 5  |-  1P  e.  P.
5 mulsrpr 9357 . . . . 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 685 . . . 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 9303 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
84, 7mpan2 671 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
9 mulclpr 9303 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
104, 9mpan2 671 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
11 addclpr 9301 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  e. 
P. )
128, 10, 11syl2an 477 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )
1312, 12anim12i 566 . . . . . . 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 2454 . . . . . . . 8  |-  ( ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  +P. 
1P )  =  ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  +P.  1P )
15 enreceq 9350 . . . . . . . 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 471 . . . . . 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 685 . . . . 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 645 . . . 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 2495 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R  )
21 df-0r 9345 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
2221oveq2i 6214 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  0R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )
2320, 22, 213eqtr4g 2520 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R )
241, 3, 23ecoptocl 7303 1  |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3994  (class class class)co 6203   [cec 7212   P.cnp 9140   1Pc1p 9141    +P. cpp 9142    .P. cmp 9143    ~R cer 9147   R.cnr 9148   0Rc0r 9149    .R cmr 9153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-omul 7038  df-er 7214  df-ec 7216  df-qs 7220  df-ni 9155  df-pli 9156  df-mi 9157  df-lti 9158  df-plpq 9191  df-mpq 9192  df-ltpq 9193  df-enq 9194  df-nq 9195  df-erq 9196  df-plq 9197  df-mq 9198  df-1nq 9199  df-rq 9200  df-ltnq 9201  df-np 9264  df-1p 9265  df-plp 9266  df-mp 9267  df-ltp 9268  df-mpr 9339  df-enr 9340  df-nr 9341  df-mr 9343  df-0r 9345
This theorem is referenced by:  pn0sr  9382  mulresr  9420  axi2m1  9440  axcnre  9445
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