Theorem 1idsr 7987

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 7946	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6024	. . 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 2246	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R <-> ( A .R 1R ) = A ) )
5		df-1r 7951	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
6	5	oveq2i 6028	. . 3 |- ( [ <. x , y >. ] ~R .R 1R ) = ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
7		1pr 7773	. . . . . 6 |- 1P e. P.
8		addclpr 7756	. . . . . 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 7965	. . . . 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 7807	. . . . . . . . 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 1365	. . . . . . . 8 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) = ( ( x .P. 1P ) +P. ( x .P. 1P ) ) )
14		1idpr 7811	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) = x )
15	14	oveq1d 6032	. . . . . . . 8 |- ( x e. P. -> ( ( x .P. 1P ) +P. ( x .P. 1P ) ) = ( x +P. ( x .P. 1P ) ) )
16	13, 15	eqtr2d 2265	. . . . . . 7 |- ( x e. P. -> ( x +P. ( x .P. 1P ) ) = ( x .P. ( 1P +P. 1P ) ) )
17		distrprg 7807	. . . . . . . . 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 1365	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( ( y .P. 1P ) +P. ( y .P. 1P ) ) )
19		1idpr 7811	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) = y )
20	19	oveq1d 6032	. . . . . . . 8 |- ( y e. P. -> ( ( y .P. 1P ) +P. ( y .P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
21	18, 20	eqtrd 2264	. . . . . . 7 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
22	16, 21	oveqan12d 6036	. . . . . 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 7791	. . . . . . . 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 7791	. . . . . . . . 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 7798	. . . . . . 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 1273	. . . . . 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 7791	. . . . . . . 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 7791	. . . . . . . 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 7797	. . . . . . . 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 7798	. . . . . . . 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 6203	. . . . . 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 2272	. . . . 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 7756	. . . . . . . . 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 7756	. . . . . . . . 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 7955	. . . . . . 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 2267	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. x , y >. ] ~R )
55	6, 54	eqtrid 2276	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R )
56	1, 4, 55	ecoptocl 6790	1 |- ( A e. R. -> ( A .R 1R ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   <.cop 3672  (class class class)co 6017   [cec 6699   P.cnp 7510   1Pc1p 7511   +P. cpp 7512   .P. cmp 7513   ~R cer 7515   R.cnr 7516   1Rc1r 7518   .R cmr 7521

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-iplp 7687  df-imp 7688  df-enr 7945  df-nr 7946  df-mr 7948  df-1r 7951

This theorem is referenced by:  pn0sr  7990  axi2m1  8094  ax1rid  8096  axcnre  8100