Theorem 1idsr 7828

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 7787	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 5925	. . 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 2208	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R <-> ( A .R 1R ) = A ) )
5		df-1r 7792	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
6	5	oveq2i 5929	. . 3 |- ( [ <. x , y >. ] ~R .R 1R ) = ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
7		1pr 7614	. . . . . 6 |- 1P e. P.
8		addclpr 7597	. . . . . 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 7806	. . . . 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 7648	. . . . . . . . 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 1340	. . . . . . . 8 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) = ( ( x .P. 1P ) +P. ( x .P. 1P ) ) )
14		1idpr 7652	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) = x )
15	14	oveq1d 5933	. . . . . . . 8 |- ( x e. P. -> ( ( x .P. 1P ) +P. ( x .P. 1P ) ) = ( x +P. ( x .P. 1P ) ) )
16	13, 15	eqtr2d 2227	. . . . . . 7 |- ( x e. P. -> ( x +P. ( x .P. 1P ) ) = ( x .P. ( 1P +P. 1P ) ) )
17		distrprg 7648	. . . . . . . . 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 1340	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( ( y .P. 1P ) +P. ( y .P. 1P ) ) )
19		1idpr 7652	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) = y )
20	19	oveq1d 5933	. . . . . . . 8 |- ( y e. P. -> ( ( y .P. 1P ) +P. ( y .P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
21	18, 20	eqtrd 2226	. . . . . . 7 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
22	16, 21	oveqan12d 5937	. . . . . 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 7632	. . . . . . . 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 7632	. . . . . . . . 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 7639	. . . . . . 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 1249	. . . . . 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 7632	. . . . . . . 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 7632	. . . . . . . 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 7638	. . . . . . . 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 7639	. . . . . . . 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 6100	. . . . . 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 2234	. . . . 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 7597	. . . . . . . . 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 7597	. . . . . . . . 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 7796	. . . . . . 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 2229	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. x , y >. ] ~R )
55	6, 54	eqtrid 2238	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R )
56	1, 4, 55	ecoptocl 6676	1 |- ( A e. R. -> ( A .R 1R ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   <.cop 3621  (class class class)co 5918   [cec 6585   P.cnp 7351   1Pc1p 7352   +P. cpp 7353   .P. cmp 7354   ~R cer 7356   R.cnr 7357   1Rc1r 7359   .R cmr 7362

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-imp 7529  df-enr 7786  df-nr 7787  df-mr 7789  df-1r 7792

This theorem is referenced by:  pn0sr  7831  axi2m1  7935  ax1rid  7937  axcnre  7941