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Theorem 00sr 8979
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 8940 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6091 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R  0R )  =  ( A  .R  0R ) )
32eqeq1d 2446 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R  <->  ( A  .R  0R )  =  0R ) )
4 1pr 8897 . . . . 5  |-  1P  e.  P.
5 mulsrpr 8956 . . . . 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 668 . . . 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 8902 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
84, 7mpan2 654 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
9 mulclpr 8902 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
104, 9mpan2 654 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
11 addclpr 8900 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  e. 
P. )
128, 10, 11syl2an 465 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  e.  P. )
1312, 12anim12i 551 . . . . . . 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 2438 . . . . . . . 8  |-  ( ( ( x  .P.  1P )  +P.  ( y  .P. 
1P ) )  +P. 
1P )  =  ( ( ( x  .P.  1P )  +P.  ( y  .P.  1P ) )  +P.  1P )
15 enreceq 8949 . . . . . . . 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 226 . . . . . . 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 459 . . . . . 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 668 . . . . 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 628 . . . 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 2470 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )  =  [ <. 1P ,  1P >. ]  ~R  )
21 df-0r 8944 . . . 4  |-  0R  =  [ <. 1P ,  1P >. ]  ~R
2221oveq2i 6095 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  0R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. 1P ,  1P >. ]  ~R  )
2320, 22, 213eqtr4g 2495 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R  0R )  =  0R )
241, 3, 23ecoptocl 6997 1  |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819  (class class class)co 6084   [cec 6906   P.cnp 8739   1Pc1p 8740    +P. cpp 8741    .P. cmp 8742    ~R cer 8746   R.cnr 8747   0Rc0r 8748    .R cmr 8752
This theorem is referenced by:  pn0sr  8981  mulresr  9019  axi2m1  9039  axcnre  9044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-ni 8754  df-pli 8755  df-mi 8756  df-lti 8757  df-plpq 8790  df-mpq 8791  df-ltpq 8792  df-enq 8793  df-nq 8794  df-erq 8795  df-plq 8796  df-mq 8797  df-1nq 8798  df-rq 8799  df-ltnq 8800  df-np 8863  df-1p 8864  df-plp 8865  df-mp 8866  df-ltp 8867  df-mpr 8938  df-enr 8939  df-nr 8940  df-mr 8942  df-0r 8944
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