Theorem 1idsr 7767

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 7726	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5882	. . 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 2192	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R <-> ( A .R 1R ) = A ) )
5		df-1r 7731	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
6	5	oveq2i 5886	. . 3 |- ( [ <. x , y >. ] ~R .R 1R ) = ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
7		1pr 7553	. . . . . 6 |- 1P e. P.
8		addclpr 7536	. . . . . 6 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
9	7, 7, 8	mp2an 426	. . . . 5 |- ( 1P +P. 1P ) e. P.
10		mulsrpr 7745	. . . . 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 439	. . . 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 7587	. . . . . . . . 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 1329	. . . . . . . 8 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) = ( ( x .P. 1P ) +P. ( x .P. 1P ) ) )
14		1idpr 7591	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) = x )
15	14	oveq1d 5890	. . . . . . . 8 |- ( x e. P. -> ( ( x .P. 1P ) +P. ( x .P. 1P ) ) = ( x +P. ( x .P. 1P ) ) )
16	13, 15	eqtr2d 2211	. . . . . . 7 |- ( x e. P. -> ( x +P. ( x .P. 1P ) ) = ( x .P. ( 1P +P. 1P ) ) )
17		distrprg 7587	. . . . . . . . 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 1329	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( ( y .P. 1P ) +P. ( y .P. 1P ) ) )
19		1idpr 7591	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) = y )
20	19	oveq1d 5890	. . . . . . . 8 |- ( y e. P. -> ( ( y .P. 1P ) +P. ( y .P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
21	18, 20	eqtrd 2210	. . . . . . 7 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
22	16, 21	oveqan12d 5894	. . . . . 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 109	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> x e. P. )
24		mulclpr 7571	. . . . . . . 8 |- ( ( x e. P. /\ 1P e. P. ) -> ( x .P. 1P ) e. P. )
25	23, 7, 24	sylancl 413	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> ( x .P. 1P ) e. P. )
26		mulclpr 7571	. . . . . . . . 9 |- ( ( y e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( y .P. ( 1P +P. 1P ) ) e. P. )
27	9, 26	mpan2 425	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) e. P. )
28	27	adantl 277	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> ( y .P. ( 1P +P. 1P ) ) e. P. )
29		addassprg 7578	. . . . . . 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 1238	. . . . . 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 7571	. . . . . . . 8 |- ( ( x e. P. /\ ( 1P +P. 1P ) e. P. ) -> ( x .P. ( 1P +P. 1P ) ) e. P. )
32	23, 9, 31	sylancl 413	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> ( x .P. ( 1P +P. 1P ) ) e. P. )
33		simpr 110	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> y e. P. )
34		mulclpr 7571	. . . . . . . 8 |- ( ( y e. P. /\ 1P e. P. ) -> ( y .P. 1P ) e. P. )
35	33, 7, 34	sylancl 413	. . . . . . 7 |- ( ( x e. P. /\ y e. P. ) -> ( y .P. 1P ) e. P. )
36		addcomprg 7577	. . . . . . . 8 |- ( ( z e. P. /\ w e. P. ) -> ( z +P. w ) = ( w +P. z ) )
37	36	adantl 277	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( z +P. w ) = ( w +P. z ) )
38		addassprg 7578	. . . . . . . 8 |- ( ( z e. P. /\ w e. P. /\ v e. P. ) -> ( ( z +P. w ) +P. v ) = ( z +P. ( w +P. v ) ) )
39	38	adantl 277	. . . . . . 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 6056	. . . . . 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 2218	. . . . 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 425	. . . . . . . . 9 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) e. P. )
43	7, 34	mpan2 425	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) e. P. )
44		addclpr 7536	. . . . . . . . 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 289	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. ( 1P +P. 1P ) ) +P. ( y .P. 1P ) ) e. P. )
46	7, 24	mpan2 425	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) e. P. )
47		addclpr 7536	. . . . . . . . 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 289	. . . . . . . 8 |- ( ( x e. P. /\ y e. P. ) -> ( ( x .P. 1P ) +P. ( y .P. ( 1P +P. 1P ) ) ) e. P. )
49	45, 48	anim12i 338	. . . . . . 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 7735	. . . . . . 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 282	. . . . . 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 397	. . . . 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 167	. . . 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 2213	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. x , y >. ] ~R )
55	6, 54	eqtrid 2222	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R )
56	1, 4, 55	ecoptocl 6622	1 |- ( A e. R. -> ( A .R 1R ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   <.cop 3596  (class class class)co 5875   [cec 6533   P.cnp 7290   1Pc1p 7291   +P. cpp 7292   .P. cmp 7293   ~R cer 7295   R.cnr 7296   1Rc1r 7298   .R cmr 7301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-2o 6418  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-pli 7304  df-mi 7305  df-lti 7306  df-plpq 7343  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-plqqs 7348  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-enq0 7423  df-nq0 7424  df-0nq0 7425  df-plq0 7426  df-mq0 7427  df-inp 7465  df-i1p 7466  df-iplp 7467  df-imp 7468  df-enr 7725  df-nr 7726  df-mr 7728  df-1r 7731

This theorem is referenced by:  pn0sr  7770  axi2m1  7874  ax1rid  7876  axcnre  7880