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Theorem 1idsr 9270
Description: 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
1idsr  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )

Proof of Theorem 1idsr
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nr 9232 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6103 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
1R )  =  ( A  .R  1R )
)
3 id 22 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  ->  [ <. x ,  y
>. ]  ~R  =  A )
42, 3eqeq12d 2457 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  .R  1R )  =  A
) )
5 df-1r 9237 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
65oveq2i 6107 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  1R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
7 1pr 9189 . . . . . 6  |-  1P  e.  P.
8 addclpr 9192 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
97, 7, 8mp2an 672 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
10 mulsrpr 9248 . . . . 5  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( 1P  +P.  1P )  e.  P.  /\  1P  e.  P. ) )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  )
119, 7, 10mpanr12 685 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  )
12 distrpr 9202 . . . . . . . 8  |-  ( x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) )
13 1idpr 9203 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  =  x )
1413oveq1d 6111 . . . . . . . 8  |-  ( x  e.  P.  ->  (
( x  .P.  1P )  +P.  ( x  .P.  1P ) )  =  ( x  +P.  ( x  .P.  1P ) ) )
1512, 14syl5req 2488 . . . . . . 7  |-  ( x  e.  P.  ->  (
x  +P.  ( x  .P.  1P ) )  =  ( x  .P.  ( 1P  +P.  1P ) ) )
16 distrpr 9202 . . . . . . . 8  |-  ( y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P.  1P )  +P.  ( y  .P. 
1P ) )
17 1idpr 9203 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  =  y )
1817oveq1d 6111 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( y  .P.  1P )  +P.  ( y  .P. 
1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
1916, 18syl5eq 2487 . . . . . . 7  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2015, 19oveqan12d 6115 . . . . . 6  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  ( x  .P.  1P ) )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  =  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  +P.  ( y  .P. 
1P ) ) ) )
21 addasspr 9196 . . . . . 6  |-  ( ( x  +P.  ( x  .P.  1P ) )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  =  ( x  +P.  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) )
22 ovex 6121 . . . . . . 7  |-  ( x  .P.  ( 1P  +P.  1P ) )  e.  _V
23 vex 2980 . . . . . . 7  |-  y  e. 
_V
24 ovex 6121 . . . . . . 7  |-  ( y  .P.  1P )  e. 
_V
25 addcompr 9195 . . . . . . 7  |-  ( z  +P.  w )  =  ( w  +P.  z
)
26 addasspr 9196 . . . . . . 7  |-  ( ( z  +P.  w )  +P.  v )  =  ( z  +P.  (
w  +P.  v )
)
2722, 23, 24, 25, 26caov12 6296 . . . . . 6  |-  ( ( x  .P.  ( 1P 
+P.  1P ) )  +P.  ( y  +P.  (
y  .P.  1P )
) )  =  ( y  +P.  ( ( x  .P.  ( 1P 
+P.  1P ) )  +P.  ( y  .P.  1P ) ) )
2820, 21, 273eqtr3g 2498 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  +P.  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) )  =  ( y  +P.  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ) )
29 mulclpr 9194 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
309, 29mpan2 671 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  e. 
P. )
31 mulclpr 9194 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
327, 31mpan2 671 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
33 addclpr 9192 . . . . . . . . 9  |-  ( ( ( x  .P.  ( 1P  +P.  1P ) )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P. 
1P ) )  e. 
P. )
3430, 32, 33syl2an 477 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) )  e.  P. )
35 mulclpr 9194 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
367, 35mpan2 671 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
37 mulclpr 9194 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
389, 37mpan2 671 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )
39 addclpr 9192 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4036, 38, 39syl2an 477 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4134, 40anim12i 566 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( (
( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P. 
1P ) )  e. 
P.  /\  ( (
x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e.  P. )
)
42 enreceq 9241 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  (
y  .P.  1P )
)  e.  P.  /\  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. ) )  -> 
( [ <. x ,  y >. ]  ~R  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  (
y  .P.  1P )
) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  <->  ( x  +P.  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) )  =  ( y  +P.  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ) ) )
4341, 42syldan 470 . . . . . 6  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( x  e.  P.  /\  y  e.  P. )
)  ->  ( [ <. x ,  y >. ]  ~R  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P. 
1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  <->  ( x  +P.  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) )  =  ( y  +P.  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ) ) )
4443anidms 645 . . . . 5  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  (
y  .P.  1P )
) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  <->  ( x  +P.  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) )  =  ( y  +P.  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ) ) )
4528, 44mpbird 232 . . . 4  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  [ <. x ,  y
>. ]  ~R  =  [ <. ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ,  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) >. ]  ~R  )
4611, 45eqtr4d 2478 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. x ,  y >. ]  ~R  )
476, 46syl5eq 2487 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  )
481, 4, 47ecoptocl 7195 1  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3888  (class class class)co 6096   [cec 7104   P.cnp 9031   1Pc1p 9032    +P. cpp 9033    .P. cmp 9034    ~R cer 9038   R.cnr 9039   1Rc1r 9041    .R cmr 9044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-omul 6930  df-er 7106  df-ec 7108  df-qs 7112  df-ni 9046  df-pli 9047  df-mi 9048  df-lti 9049  df-plpq 9082  df-mpq 9083  df-ltpq 9084  df-enq 9085  df-nq 9086  df-erq 9087  df-plq 9088  df-mq 9089  df-1nq 9090  df-rq 9091  df-ltnq 9092  df-np 9155  df-1p 9156  df-plp 9157  df-mp 9158  df-ltp 9159  df-mpr 9230  df-enr 9231  df-nr 9232  df-mr 9234  df-1r 9237
This theorem is referenced by:  pn0sr  9273  sqgt0sr  9278  axi2m1  9331  ax1rid  9333  axcnre  9336
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