Theorem 1idsr 4001

Description: 1 is an identity element for multiplication.

Assertion

Ref	Expression
1idsr	|- (A e. R. -> (A .R 1R) = A)

Proof of Theorem 1idsr

Step	Hyp	Ref	Expression
1		df-nr 3961	. 2 |- R. = ((P. X. P.)/. ~R )
2		opreq1 3006	. . 3 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R .R 1R) = (A .R 1R))
3		id 9	. . 3 |- ([<.x, y>.] ~R = A -> [<.x, y>.] ~R = A)
4	2, 3	cleq12d 1115	. 2 |- ([<.x, y>.] ~R = A -> (([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R <-> (A .R 1R) = A))
5		1pr 3911	. . . . . . 7 |- 1P e. P.
6		addclpr 3914	. . . . . . 7 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
7	5, 5, 6	mp2an 520	. . . . . 6 |- (1P +P. 1P) e. P.
8	7, 5	pm3.2i 234	. . . . 5 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
9		mulsrpr 3979	. . . . 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 )
10	8, 9	mpan2 519	. . . 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 )
11		1idpr 3927	. . . . . . . . 9 |- (x e. P. -> (x .P 1P) = x)
12	11	opreq1d 3012	. . . . . . . 8 |- (x e. P. -> ((x .P 1P) +P. (x .P 1P)) = (x +P. (x .P 1P)))
13	5	elisseti 1355	. . . . . . . . 9 |- 1P e. V
14	13, 13	distrpr 3926	. . . . . . . 8 |- (x .P (1P +P. 1P)) = ((x .P 1P) +P. (x .P 1P))
15	12, 14	syl5req 1137	. . . . . . 7 |- (x e. P. -> (x +P. (x .P 1P)) = (x .P (1P +P. 1P)))
16		1idpr 3927	. . . . . . . . 9 |- (y e. P. -> (y .P 1P) = y)
17	16	opreq1d 3012	. . . . . . . 8 |- (y e. P. -> ((y .P 1P) +P. (y .P 1P)) = (y +P. (y .P 1P)))
18	13, 13	distrpr 3926	. . . . . . . 8 |- (y .P (1P +P. 1P)) = ((y .P 1P) +P. (y .P 1P))
19	17, 18	syl5eq 1136	. . . . . . 7 |- (y e. P. -> (y .P (1P +P. 1P)) = (y +P. (y .P 1P)))
20	15, 19	opreqan12d 3015	. . . . . 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))))
21		oprex 3018	. . . . . . 7 |- (x .P 1P) e. V
22		oprex 3018	. . . . . . 7 |- (y .P (1P +P. 1P)) e. V
23	21, 22	addasspr 3918	. . . . . 6 |- ((x +P. (x .P 1P)) +P. (y .P (1P +P. 1P))) = (x +P. ((x .P 1P) +P. (y .P (1P +P. 1P))))
24		oprex 3018	. . . . . . 7 |- (x .P (1P +P. 1P)) e. V
25		visset 1350	. . . . . . 7 |- y e. V
26		oprex 3018	. . . . . . 7 |- (y .P 1P) e. V
27		visset 1350	. . . . . . . 8 |- z e. V
28		visset 1350	. . . . . . . 8 |- w e. V
29	27, 28	addcompr 3917	. . . . . . 7 |- (z +P. w) = (w +P. z)
30		visset 1350	. . . . . . . 8 |- v e. V
31	28, 30	addasspr 3918	. . . . . . 7 |- ((z +P. w) +P. v) = (z +P. (w +P. v))
32	24, 25, 26, 29, 31	caopr12 3075	. . . . . 6 |- ((x .P (1P +P. 1P)) +P. (y +P. (y .P 1P))) = (y +P. ((x .P (1P +P. 1P)) +P. (y .P 1P)))
33	20, 23, 32	3eqtr3g 1146	. . . . 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))))
34		enreceq 3971	. . . . . . . 8 |- (((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)))))
35		addclpr 3914	. . . . . . . . . 10 |- (((x .P (1P +P. 1P)) e. P. /\ (y .P 1P) e. P.) -> ((x .P (1P +P. 1P)) +P. (y .P 1P)) e. P.)
36		mulclpr 3916	. . . . . . . . . . 11 |- ((x e. P. /\ (1P +P. 1P) e. P.) -> (x .P (1P +P. 1P)) e. P.)
37	7, 36	mpan2 519	. . . . . . . . . 10 |- (x e. P. -> (x .P (1P +P. 1P)) e. P.)
38		mulclpr 3916	. . . . . . . . . . 11 |- ((y e. P. /\ 1P e. P.) -> (y .P 1P) e. P.)
39	5, 38	mpan2 519	. . . . . . . . . 10 |- (y e. P. -> (y .P 1P) e. P.)
40	35, 37, 39	syl2an 349	. . . . . . . . 9 |- ((x e. P. /\ y e. P.) -> ((x .P (1P +P. 1P)) +P. (y .P 1P)) e. P.)
41		addclpr 3914	. . . . . . . . . 10 |- (((x .P 1P) e. P. /\ (y .P (1P +P. 1P)) e. P.) -> ((x .P 1P) +P. (y .P (1P +P. 1P))) e. P.)
42		mulclpr 3916	. . . . . . . . . . 11 |- ((x e. P. /\ 1P e. P.) -> (x .P 1P) e. P.)
43	5, 42	mpan2 519	. . . . . . . . . 10 |- (x e. P. -> (x .P 1P) e. P.)
44		mulclpr 3916	. . . . . . . . . . 11 |- ((y e. P. /\ (1P +P. 1P) e. P.) -> (y .P (1P +P. 1P)) e. P.)
45	7, 44	mpan2 519	. . . . . . . . . 10 |- (y e. P. -> (y .P (1P +P. 1P)) e. P.)
46	41, 43, 45	syl2an 349	. . . . . . . . 9 |- ((x e. P. /\ y e. P.) -> ((x .P 1P) +P. (y .P (1P +P. 1P))) e. P.)
47	40, 46	anim12i 268	. . . . . . . 8 |- (((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.))
48	34, 47	sylan2 346	. . . . . . 7 |- (((x e. P. /\ y e. P.) /\ ((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)))))
49	48	anabss5 384	. . . . . 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)))))
50	49	anidms 332	. . . . 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)))))
51	33, 50	mpbird 171	. . . 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 )
52	10, 51	eqtr4d 1131	. . 3 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R ) = [<.x, y>.] ~R )
53		df-1r 3966	. . . 4 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
54	53	opreq2i 3010	. . 3 |- ([<.x, y>.] ~R .R 1R) = ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R )
55	52, 54	syl5eq 1136	. 2 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R )
56	1, 4, 55	ecoptocl 3239	1 |- (A e. R. -> (A .R 1R) = A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  <.cop 1810  (class class class)co 3001  [cec 3198  P.cnp 3779  1Pc1p 3780   +P. cpp 3781   .P cmp 3782   ~R cer 3786  R.cnr 3787  1Rc1r 3789   .R cmr 3792

This theorem is referenced by:  pn0sr 4004  sqgt0sr 4009  ax1id 4077  axrecex 4079  axi2m1 4082  axcnre 4087

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr