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Theorem 1idsr 9471
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 9430 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 6289 . . 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 2489 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  .R  1R )  =  A
) )
5 df-1r 9435 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
65oveq2i 6293 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  1R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
7 1pr 9389 . . . . . 6  |-  1P  e.  P.
8 addclpr 9392 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
97, 7, 8mp2an 672 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
10 mulsrpr 9449 . . . . 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 9402 . . . . . . . 8  |-  ( x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) )
13 1idpr 9403 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  =  x )
1413oveq1d 6297 . . . . . . . 8  |-  ( x  e.  P.  ->  (
( x  .P.  1P )  +P.  ( x  .P.  1P ) )  =  ( x  +P.  ( x  .P.  1P ) ) )
1512, 14syl5req 2521 . . . . . . 7  |-  ( x  e.  P.  ->  (
x  +P.  ( x  .P.  1P ) )  =  ( x  .P.  ( 1P  +P.  1P ) ) )
16 distrpr 9402 . . . . . . . 8  |-  ( y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P.  1P )  +P.  ( y  .P. 
1P ) )
17 1idpr 9403 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  =  y )
1817oveq1d 6297 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( y  .P.  1P )  +P.  ( y  .P. 
1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
1916, 18syl5eq 2520 . . . . . . 7  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2015, 19oveqan12d 6301 . . . . . 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 9396 . . . . . 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 6307 . . . . . . 7  |-  ( x  .P.  ( 1P  +P.  1P ) )  e.  _V
23 vex 3116 . . . . . . 7  |-  y  e. 
_V
24 ovex 6307 . . . . . . 7  |-  ( y  .P.  1P )  e. 
_V
25 addcompr 9395 . . . . . . 7  |-  ( z  +P.  w )  =  ( w  +P.  z
)
26 addasspr 9396 . . . . . . 7  |-  ( ( z  +P.  w )  +P.  v )  =  ( z  +P.  (
w  +P.  v )
)
2722, 23, 24, 25, 26caov12 6485 . . . . . 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 2531 . . . . 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 9394 . . . . . . . . . 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 9394 . . . . . . . . . 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 9392 . . . . . . . . 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 9394 . . . . . . . . . 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 9394 . . . . . . . . . 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 9392 . . . . . . . . 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 9439 . . . . . . 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 2511 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. x ,  y >. ]  ~R  )
476, 46syl5eq 2520 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  )
481, 4, 47ecoptocl 7398 1  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033  (class class class)co 6282   [cec 7306   P.cnp 9233   1Pc1p 9234    +P. cpp 9235    .P. cmp 9236    ~R cer 9238   R.cnr 9239   1Rc1r 9241    .R cmr 9244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-ni 9246  df-pli 9247  df-mi 9248  df-lti 9249  df-plpq 9282  df-mpq 9283  df-ltpq 9284  df-enq 9285  df-nq 9286  df-erq 9287  df-plq 9288  df-mq 9289  df-1nq 9290  df-rq 9291  df-ltnq 9292  df-np 9355  df-1p 9356  df-plp 9357  df-mp 9358  df-ltp 9359  df-enr 9429  df-nr 9430  df-mr 9432  df-1r 9435
This theorem is referenced by:  pn0sr  9474  sqgt0sr  9479  axi2m1  9532  ax1rid  9534  axcnre  9537
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