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.

Step	Hyp	Ref	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 )
4	2, 3	eqeq12d 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
6	5	oveq2i 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. )
9	7, 7, 8	mp2an 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 )
11	9, 7, 10	mpanr12 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 ) ) )
13	7, 7, 12	mp3an23 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 )
15	14	oveq1d 5578	. . . . . . . 8 |- ( x e. P. -> ( ( x .P. 1P ) +P. ( x .P. 1P ) ) = ( x +P. ( x .P. 1P ) ) )
16	13, 15	eqtr2d 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 ) ) )
18	7, 7, 17	mp3an23 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 )
20	19	oveq1d 5578	. . . . . . . 8 |- ( y e. P. -> ( ( y .P. 1P ) +P. ( y .P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
21	18, 20	eqtrd 2115	. . . . . . 7 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
22	16, 21	oveqan12d 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. )
25	23, 7, 24	sylancl 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. )
27	9, 26	mpan2 416	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) e. P. )
28	27	adantl 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 ) ) ) ) )
30	23, 25, 28, 29	syl3anc 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. )
32	23, 9, 31	sylancl 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. )
35	33, 7, 34	sylancl 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 ) )
37	36	adantl 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 ) ) )
39	38	adantl 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 ) ) )
40	32, 33, 35, 37, 39	caov12d 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 ) ) ) )
41	22, 30, 40	3eqtr3d 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 ) ) ) )
42	9, 31	mpan2 416	. . . . . . . . 9 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) e. P. )
43	7, 34	mpan2 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. )
45	42, 43, 44	syl2an 283	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. ( 1P +P. 1P ) ) +P. ( y .P. 1P ) ) e. P. )
46	7, 24	mpan2 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. )
48	46, 27, 47	syl2an 283	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. ( 1P +P. 1P ) ) ) e. P. )
49	45, 48	anim12i 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 ) ) ) ) )
51	49, 50	syldan 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 ) ) ) ) )
52	51	anidms 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 ) ) ) ) )
53	41, 52	mpbird 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 )
54	11, 53	eqtr4d 2118	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. x , y >. ] ~R )
55	6, 54	syl5eq 2127	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R )
56	1, 4, 55	ecoptocl 6280	1 |- ( A e. R. -> ( A .R 1R ) = A )

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