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Theorem 1idsr 6910
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 6869 . 2  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
2 oveq1 5546 . . 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 2070 . 2  |-  ( [
<. x ,  y >. ]  ~R  =  A  -> 
( ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  <->  ( A  .R  1R )  =  A
) )
5 df-1r 6874 . . . 4  |-  1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
65oveq2i 5550 . . 3  |-  ( [
<. x ,  y >. ]  ~R  .R  1R )  =  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )
7 1pr 6709 . . . . . 6  |-  1P  e.  P.
8 addclpr 6692 . . . . . 6  |-  ( ( 1P  e.  P.  /\  1P  e.  P. )  -> 
( 1P  +P.  1P )  e.  P. )
97, 7, 8mp2an 410 . . . . 5  |-  ( 1P 
+P.  1P )  e.  P.
10 mulsrpr 6888 . . . . 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 423 . . . 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 6743 . . . . . . . . 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 1235 . . . . . . . 8  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  =  ( ( x  .P.  1P )  +P.  ( x  .P.  1P ) ) )
14 1idpr 6747 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  =  x )
1514oveq1d 5554 . . . . . . . 8  |-  ( x  e.  P.  ->  (
( x  .P.  1P )  +P.  ( x  .P.  1P ) )  =  ( x  +P.  ( x  .P.  1P ) ) )
1613, 15eqtr2d 2089 . . . . . . 7  |-  ( x  e.  P.  ->  (
x  +P.  ( x  .P.  1P ) )  =  ( x  .P.  ( 1P  +P.  1P ) ) )
17 distrprg 6743 . . . . . . . . 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 1235 . . . . . . . 8  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( ( y  .P. 
1P )  +P.  (
y  .P.  1P )
) )
19 1idpr 6747 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  =  y )
2019oveq1d 5554 . . . . . . . 8  |-  ( y  e.  P.  ->  (
( y  .P.  1P )  +P.  ( y  .P. 
1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2118, 20eqtrd 2088 . . . . . . 7  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  =  ( y  +P.  (
y  .P.  1P )
) )
2216, 21oveqan12d 5558 . . . . . 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 106 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  x  e.  P. )
24 mulclpr 6727 . . . . . . . 8  |-  ( ( x  e.  P.  /\  1P  e.  P. )  -> 
( x  .P.  1P )  e.  P. )
2523, 7, 24sylancl 398 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  .P.  1P )  e.  P. )
26 mulclpr 6727 . . . . . . . . 9  |-  ( ( y  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
279, 26mpan2 409 . . . . . . . 8  |-  ( y  e.  P.  ->  (
y  .P.  ( 1P  +P.  1P ) )  e. 
P. )
2827adantl 266 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( y  .P.  ( 1P  +P.  1P ) )  e.  P. )
29 addassprg 6734 . . . . . . 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 1146 . . . . . 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 6727 . . . . . . . 8  |-  ( ( x  e.  P.  /\  ( 1P  +P.  1P )  e.  P. )  -> 
( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
3223, 9, 31sylancl 398 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( x  .P.  ( 1P  +P.  1P ) )  e.  P. )
33 simpr 107 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  y  e.  P. )
34 mulclpr 6727 . . . . . . . 8  |-  ( ( y  e.  P.  /\  1P  e.  P. )  -> 
( y  .P.  1P )  e.  P. )
3533, 7, 34sylancl 398 . . . . . . 7  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( y  .P.  1P )  e.  P. )
36 addcomprg 6733 . . . . . . . 8  |-  ( ( z  e.  P.  /\  w  e.  P. )  ->  ( z  +P.  w
)  =  ( w  +P.  z ) )
3736adantl 266 . . . . . . 7  |-  ( ( ( x  e.  P.  /\  y  e.  P. )  /\  ( z  e.  P.  /\  w  e.  P. )
)  ->  ( z  +P.  w )  =  ( w  +P.  z ) )
38 addassprg 6734 . . . . . . . 8  |-  ( ( z  e.  P.  /\  w  e.  P.  /\  v  e.  P. )  ->  (
( z  +P.  w
)  +P.  v )  =  ( z  +P.  ( w  +P.  v
) ) )
3938adantl 266 . . . . . . 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 5709 . . . . . 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 2096 . . . . 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 409 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  ( 1P  +P.  1P ) )  e. 
P. )
437, 34mpan2 409 . . . . . . . . 9  |-  ( y  e.  P.  ->  (
y  .P.  1P )  e.  P. )
44 addclpr 6692 . . . . . . . . 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 277 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  ( 1P  +P.  1P ) )  +P.  ( y  .P.  1P ) )  e.  P. )
467, 24mpan2 409 . . . . . . . . 9  |-  ( x  e.  P.  ->  (
x  .P.  1P )  e.  P. )
47 addclpr 6692 . . . . . . . . 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 277 . . . . . . . 8  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( ( x  .P.  1P )  +P.  ( y  .P.  ( 1P  +P.  1P ) ) )  e. 
P. )
4945, 48anim12i 325 . . . . . . 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 6878 . . . . . . 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 270 . . . . . 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 383 . . . . 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 160 . . . 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 2091 . . 3  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
[ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R  )  =  [ <. x ,  y >. ]  ~R  )
556, 54syl5eq 2100 . 2  |-  ( ( x  e.  P.  /\  y  e.  P. )  ->  ( [ <. x ,  y >. ]  ~R  .R 
1R )  =  [ <. x ,  y >. ]  ~R  )
561, 4, 55ecoptocl 6223 1  |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   <.cop 3405  (class class class)co 5539   [cec 6134   P.cnp 6446   1Pc1p 6447    +P. cpp 6448    .P. cmp 6449    ~R cer 6451   R.cnr 6452   1Rc1r 6454    .R cmr 6457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-imp 6624  df-enr 6868  df-nr 6869  df-mr 6871  df-1r 6874
This theorem is referenced by:  pn0sr  6913  axi2m1  7006  ax1rid  7008  axcnre  7012
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