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Theorem 1idsr 7059
Description: 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
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 7018 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5570 . . 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 2097 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  .R  1R )  =  A
) )
5 df-1r 7023 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
65oveq2i 5574 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  1R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
7 1pr 6858 . . . . . 6  |-  1P  e.  P.
8 addclpr 6841 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
97, 7, 8mp2an 417 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
10 mulsrpr 7037 . . . . 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 430 . . . 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 distrprg 6892 . . . . . . . . 9  |-  ( ( x  e.  P.  /\  1P  e.  P.  /\  1P  e.  P. )  ->  (
x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) ) )
137, 7, 12mp3an23 1261 . . . . . . . 8  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) ) )
14 1idpr 6896 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  =  x )
1514oveq1d 5578 . . . . . . . 8  |-  ( x  e.  P.  ->  (
( x  .P.  1P )  +P.  ( x  .P.  1P ) )  =  ( x  +P.  ( x  .P.  1P ) ) )
1613, 15eqtr2d 2116 . . . . . . 7  |-  ( x  e.  P.  ->  (
x  +P.  ( x  .P.  1P ) )  =  ( x  .P.  ( 1P  +P.  1P ) ) )
17 distrprg 6892 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  1P  e.  P.  /\  1P  e.  P. )  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P. 
1P )  +P.  (
y  .P.  1P )
) )
187, 7, 17mp3an23 1261 . . . . . . . 8  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P. 
1P )  +P.  (
y  .P.  1P )
) )
19 1idpr 6896 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  =  y )
2019oveq1d 5578 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( y  .P.  1P )  +P.  ( y  .P. 
1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2118, 20eqtrd 2115 . . . . . . 7  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2216, 21oveqan12d 5582 . . . . . 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 ) ) ) )
23 simpl 107 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  x  e.  P. )
24 mulclpr 6876 . . . . . . . 8  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
2523, 7, 24sylancl 404 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  .P.  1P )  e.  P. )
26 mulclpr 6876 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
279, 26mpan2 416 . . . . . . . 8  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )
2827adantl 271 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
29 addassprg 6883 . . . . . . 7  |-  ( ( x  e.  P.  /\  ( x  .P.  1P )  e.  P.  /\  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )  ->  (
( x  +P.  (
x  .P.  1P )
)  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  =  ( x  +P.  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) ) )
3023, 25, 28, 29syl3anc 1170 . . . . . 6  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  +P.  ( x  .P.  1P ) )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  =  ( x  +P.  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) ) ) )
31 mulclpr 6876 . . . . . . . 8  |-  ( ( x  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
3223, 9, 31sylancl 404 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
33 simpr 108 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  y  e.  P. )
34 mulclpr 6876 . . . . . . . 8  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
3533, 7, 34sylancl 404 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( y  .P.  1P )  e.  P. )
36 addcomprg 6882 . . . . . . . 8  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  ( z  +P.  w
)  =  ( w  +P.  z ) )
3736adantl 271 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( z  +P.  w )  =  ( w  +P.  z ) )
38 addassprg 6883 . . . . . . . 8  |-  ( ( z  e.  P.  /\  w  e.  P.  /\  v  e.  P. )  ->  (
( z  +P.  w
)  +P.  v )  =  ( z  +P.  ( w  +P.  v
) ) )
3938adantl 271 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P.  /\  v  e.  P. )
)  ->  ( (
z  +P.  w )  +P.  v )  =  ( z  +P.  ( w  +P.  v ) ) )
4032, 33, 35, 37, 39caov12d 5733 . . . . . 6  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  +P.  ( y  .P. 
1P ) ) )  =  ( y  +P.  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) ) ) )
4122, 30, 403eqtr3d 2123 . . . . 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 ) ) ) )
429, 31mpan2 416 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  e. 
P. )
437, 34mpan2 416 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
44 addclpr 6841 . . . . . . . . 9  |-  ( ( ( x  .P.  ( 1P  +P.  1P ) )  e.  P.  /\  (
y  .P.  1P )  e.  P. )  ->  (
( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P. 
1P ) )  e. 
P. )
4542, 43, 44syl2an 283 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) )  e.  P. )
467, 24mpan2 416 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
47 addclpr 6841 . . . . . . . . 9  |-  ( ( ( x  .P.  1P )  e.  P.  /\  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )  ->  (
( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4846, 27, 47syl2an 283 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4945, 48anim12i 331 . . . . . . 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. )
)
50 enreceq 7027 . . . . . . 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 ) ) ) ) )
5149, 50syldan 276 . . . . . 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 ) ) ) ) )
5251anidms 389 . . . . 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 ) ) ) ) )
5341, 52mpbird 165 . . . 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  )
5411, 53eqtr4d 2118 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. x ,  y >. ]  ~R  )
556, 54syl5eq 2127 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  )
561, 4, 55ecoptocl 6280 1  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3419  (class class class)co 5563   [cec 6191   P.cnp 6595   1Pc1p 6596    +P. cpp 6597    .P. cmp 6598    ~R cer 6600   R.cnr 6601   1Rc1r 6603    .R cmr 6606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-1o 6085  df-2o 6086  df-oadd 6089  df-omul 6090  df-er 6193  df-ec 6195  df-qs 6199  df-ni 6608  df-pli 6609  df-mi 6610  df-lti 6611  df-plpq 6648  df-mpq 6649  df-enq 6651  df-nqqs 6652  df-plqqs 6653  df-mqqs 6654  df-1nqqs 6655  df-rq 6656  df-ltnqqs 6657  df-enq0 6728  df-nq0 6729  df-0nq0 6730  df-plq0 6731  df-mq0 6732  df-inp 6770  df-i1p 6771  df-iplp 6772  df-imp 6773  df-enr 7017  df-nr 7018  df-mr 7020  df-1r 7023
This theorem is referenced by:  pn0sr  7062  axi2m1  7155  ax1rid  7157  axcnre  7161
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