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Theorem 1idsr 8736
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 8698 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5881 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( [ <. x ,  y >. ]  ~R  .R 
1R )  =  ( A  .R  1R )
)
3 id 19 . . 3  |-  ( [
<. x ,  y >. ]  ~R  =  A  ->  [ <. x ,  y
>. ]  ~R  =  A )
42, 3eqeq12d 2310 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  .R  1R )  =  A
) )
5 df-1r 8703 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
65oveq2i 5885 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  1R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
7 1pr 8655 . . . . . 6  |-  1P  e.  P.
8 addclpr 8658 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
97, 7, 8mp2an 653 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
10 mulsrpr 8714 . . . . 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 666 . . . 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 8668 . . . . . . . 8  |-  ( x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) )
13 1idpr 8669 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  =  x )
1413oveq1d 5889 . . . . . . . 8  |-  ( x  e.  P.  ->  (
( x  .P.  1P )  +P.  ( x  .P.  1P ) )  =  ( x  +P.  ( x  .P.  1P ) ) )
1512, 14syl5req 2341 . . . . . . 7  |-  ( x  e.  P.  ->  (
x  +P.  ( x  .P.  1P ) )  =  ( x  .P.  ( 1P  +P.  1P ) ) )
16 distrpr 8668 . . . . . . . 8  |-  ( y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P.  1P )  +P.  ( y  .P. 
1P ) )
17 1idpr 8669 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  =  y )
1817oveq1d 5889 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( y  .P.  1P )  +P.  ( y  .P. 
1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
1916, 18syl5eq 2340 . . . . . . 7  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2015, 19oveqan12d 5893 . . . . . 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 8662 . . . . . 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 5899 . . . . . . 7  |-  ( x  .P.  ( 1P  +P.  1P ) )  e.  _V
23 vex 2804 . . . . . . 7  |-  y  e. 
_V
24 ovex 5899 . . . . . . 7  |-  ( y  .P.  1P )  e. 
_V
25 addcompr 8661 . . . . . . 7  |-  ( z  +P.  w )  =  ( w  +P.  z
)
26 addasspr 8662 . . . . . . 7  |-  ( ( z  +P.  w )  +P.  v )  =  ( z  +P.  (
w  +P.  v )
)
2722, 23, 24, 25, 26caov12 6064 . . . . . 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 2351 . . . . 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 8660 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
309, 29mpan2 652 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  e. 
P. )
31 mulclpr 8660 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
327, 31mpan2 652 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
33 addclpr 8658 . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) )  e.  P. )
35 mulclpr 8660 . . . . . . . . . 10  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
367, 35mpan2 652 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
37 mulclpr 8660 . . . . . . . . . 10  |-  ( ( y  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
389, 37mpan2 652 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )
39 addclpr 8658 . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4134, 40anim12i 549 . . . . . . 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 8707 . . . . . . 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 456 . . . . . 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 626 . . . . 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 223 . . . 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 2331 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. x ,  y >. ]  ~R  )
476, 46syl5eq 2340 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  )
481, 4, 47ecoptocl 6764 1  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656  (class class class)co 5874   [cec 6674   P.cnp 8497   1Pc1p 8498    +P. cpp 8499    .P. cmp 8500    ~R cer 8504   R.cnr 8505   1Rc1r 8507    .R cmr 8510
This theorem is referenced by:  pn0sr  8739  sqgt0sr  8744  axi2m1  8797  ax1rid  8799  axcnre  8802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-plp 8623  df-mp 8624  df-ltp 8625  df-mpr 8696  df-enr 8697  df-nr 8698  df-mr 8700  df-1r 8703
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