Theorem 1idsr 8083

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 8042	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6057	. . 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 2247	. 2 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R <-> ( A .R 1R ) = A ) )
5		df-1r 8047	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
6	5	oveq2i 6061	. . 3 |- ( [ <. x , y >. ] ~R .R 1R ) = ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
7		1pr 7869	. . . . . 6 |- 1P e. P.
8		addclpr 7852	. . . . . 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 8061	. . . . 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 7903	. . . . . . . . 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 1366	. . . . . . . 8 |- ( x e. P. -> ( x .P. ( 1P +P. 1P ) ) = ( ( x .P. 1P ) +P. ( x .P. 1P ) ) )
14		1idpr 7907	. . . . . . . . 9 |- ( x e. P. -> ( x .P. 1P ) = x )
15	14	oveq1d 6065	. . . . . . . 8 |- ( x e. P. -> ( ( x .P. 1P ) +P. ( x .P. 1P ) ) = ( x +P. ( x .P. 1P ) ) )
16	13, 15	eqtr2d 2266	. . . . . . 7 |- ( x e. P. -> ( x +P. ( x .P. 1P ) ) = ( x .P. ( 1P +P. 1P ) ) )
17		distrprg 7903	. . . . . . . . 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 1366	. . . . . . . 8 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( ( y .P. 1P ) +P. ( y .P. 1P ) ) )
19		1idpr 7907	. . . . . . . . 9 |- ( y e. P. -> ( y .P. 1P ) = y )
20	19	oveq1d 6065	. . . . . . . 8 |- ( y e. P. -> ( ( y .P. 1P ) +P. ( y .P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
21	18, 20	eqtrd 2265	. . . . . . 7 |- ( y e. P. -> ( y .P. ( 1P +P. 1P ) ) = ( y +P. ( y .P. 1P ) ) )
22	16, 21	oveqan12d 6069	. . . . . 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 7887	. . . . . . . 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 7887	. . . . . . . . 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 7894	. . . . . . 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 1274	. . . . . 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 7887	. . . . . . . 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 7887	. . . . . . . 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 7893	. . . . . . . 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 7894	. . . . . . . 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 6236	. . . . . 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 2273	. . . . 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 7852	. . . . . . . . 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 7852	. . . . . . . . 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 8051	. . . . . . 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 2268	. . 3 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. x , y >. ] ~R )
55	6, 54	eqtrid 2277	. 2 |- ( ( x e. P. /\ y e. P. ) -> ( [ <. x , y >. ] ~R .R 1R ) = [ <. x , y >. ] ~R )
56	1, 4, 55	ecoptocl 6856	1 |- ( A e. R. -> ( A .R 1R ) = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   <.cop 3692  (class class class)co 6050   [cec 6765   P.cnp 7606   1Pc1p 7607   +P. cpp 7608   .P. cmp 7609   ~R cer 7611   R.cnr 7612   1Rc1r 7614   .R cmr 7617

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-iplp 7783  df-imp 7784  df-enr 8041  df-nr 8042  df-mr 8044  df-1r 8047

This theorem is referenced by:  pn0sr  8086  axi2m1  8190  ax1rid  8192  axcnre  8196