Theorem 1idsr 5272

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 5232	. 2 |- R. = ((P. X. P.)/. ~R )
2		opreq1 4026	. . 3 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R .R 1R) = (A .R 1R))
3		id 59	. . 3 |- ([<.x, y>.] ~R = A -> [<.x, y>.] ~R = A)
4	2, 3	eqeq12d 1536	. 2 |- ([<.x, y>.] ~R = A -> (([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R <-> (A .R 1R) = A))
5		1pr 5182	. . . . . . 7 |- 1P e. P.
6		addclpr 5185	. . . . . . 7 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
7	5, 5, 6	mp2an 709	. . . . . 6 |- (1P +P. 1P) e. P.
8	7, 5	pm3.2i 292	. . . . 5 |- ((1P +P. 1P) e. P. /\ 1P e. P.)
9		mulsrpr 5250	. . . . 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 708	. . . 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 5198	. . . . . . . . 9 |- (x e. P. -> (x .P. 1P) = x)
12	11	opreq1d 4033	. . . . . . . 8 |- (x e. P. -> ((x .P. 1P) +P. (x .P. 1P)) = (x +P. (x .P. 1P)))
13	5	elisseti 1865	. . . . . . . . 9 |- 1P e. V
14	13, 13	distrpr 5197	. . . . . . . 8 |- (x .P. (1P +P. 1P)) = ((x .P. 1P) +P. (x .P. 1P))
15	12, 14	syl5req 1567	. . . . . . 7 |- (x e. P. -> (x +P. (x .P. 1P)) = (x .P. (1P +P. 1P)))
16		1idpr 5198	. . . . . . . . 9 |- (y e. P. -> (y .P. 1P) = y)
17	16	opreq1d 4033	. . . . . . . 8 |- (y e. P. -> ((y .P. 1P) +P. (y .P. 1P)) = (y +P. (y .P. 1P)))
18	13, 13	distrpr 5197	. . . . . . . 8 |- (y .P. (1P +P. 1P)) = ((y .P. 1P) +P. (y .P. 1P))
19	17, 18	syl5eq 1566	. . . . . . 7 |- (y e. P. -> (y .P. (1P +P. 1P)) = (y +P. (y .P. 1P)))
20	15, 19	opreqan12d 4037	. . . . . 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 4041	. . . . . . 7 |- (x .P. 1P) e. V
22		oprex 4041	. . . . . . 7 |- (y .P. (1P +P. 1P)) e. V
23	21, 22	addasspr 5189	. . . . . 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 4041	. . . . . . 7 |- (x .P. (1P +P. 1P)) e. V
25		visset 1860	. . . . . . 7 |- y e. V
26		oprex 4041	. . . . . . 7 |- (y .P. 1P) e. V
27		visset 1860	. . . . . . . 8 |- z e. V
28		visset 1860	. . . . . . . 8 |- w e. V
29	27, 28	addcompr 5188	. . . . . . 7 |- (z +P. w) = (w +P. z)
30		visset 1860	. . . . . . . 8 |- v e. V
31	28, 30	addasspr 5189	. . . . . . 7 |- ((z +P. w) +P. v) = (z +P. (w +P. v))
32	24, 25, 26, 29, 31	caopr12 4119	. . . . . 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 1577	. . . . 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		addclpr 5185	. . . . . . . . 9 |- (((x .P. (1P +P. 1P)) e. P. /\ (y .P. 1P) e. P.) -> ((x .P. (1P +P. 1P)) +P. (y .P. 1P)) e. P.)
35		mulclpr 5187	. . . . . . . . . 10 |- ((x e. P. /\ (1P +P. 1P) e. P.) -> (x .P. (1P +P. 1P)) e. P.)
36	7, 35	mpan2 708	. . . . . . . . 9 |- (x e. P. -> (x .P. (1P +P. 1P)) e. P.)
37		mulclpr 5187	. . . . . . . . . 10 |- ((y e. P. /\ 1P e. P.) -> (y .P. 1P) e. P.)
38	5, 37	mpan2 708	. . . . . . . . 9 |- (y e. P. -> (y .P. 1P) e. P.)
39	34, 36, 38	syl2an 465	. . . . . . . 8 |- ((x e. P. /\ y e. P.) -> ((x .P. (1P +P. 1P)) +P. (y .P. 1P)) e. P.)
40		addclpr 5185	. . . . . . . . 9 |- (((x .P. 1P) e. P. /\ (y .P. (1P +P. 1P)) e. P.) -> ((x .P. 1P) +P. (y .P. (1P +P. 1P))) e. P.)
41		mulclpr 5187	. . . . . . . . . 10 |- ((x e. P. /\ 1P e. P.) -> (x .P. 1P) e. P.)
42	5, 41	mpan2 708	. . . . . . . . 9 |- (x e. P. -> (x .P. 1P) e. P.)
43		mulclpr 5187	. . . . . . . . . 10 |- ((y e. P. /\ (1P +P. 1P) e. P.) -> (y .P. (1P +P. 1P)) e. P.)
44	7, 43	mpan2 708	. . . . . . . . 9 |- (y e. P. -> (y .P. (1P +P. 1P)) e. P.)
45	40, 42, 44	syl2an 465	. . . . . . . 8 |- ((x e. P. /\ y e. P.) -> ((x .P. 1P) +P. (y .P. (1P +P. 1P))) e. P.)
46	39, 45	anim12i 340	. . . . . . 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.))
47		enreceq 5242	. . . . . . 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)))))
48	46, 47	syldan 478	. . . . . 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)))))
49	48	anidms 444	. . . . 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)))))
50	33, 49	mpbird 203	. . . 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 )
51	10, 50	eqtr4d 1557	. . 3 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R ) = [<.x, y>.] ~R )
52		df-1r 5237	. . . 4 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
53	52	opreq2i 4030	. . 3 |- ([<.x, y>.] ~R .R 1R) = ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R )
54	51, 53	syl5eq 1566	. 2 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R )
55	1, 4, 54	ecoptocl 4364	1 |- (A e. R. -> (A .R 1R) = A)

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  <.cop 2463  (class class class)co 4021  [cec 4317  P.cnp 5050  1Pc1p 5051   +P. cpp 5052   .P. cmp 5053   ~R cer 5057  R.cnr 5058  1Rc1r 5060   .R cmr 5063

This theorem is referenced by:  pn0sr 5275  sqgt0sr 5280  ax1id 5347  axi2m1 5350  axcnre 5351

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922  ax-inf2 4687

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-reu 1698  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-pss 2106  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-int 2588  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-om 3189  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-f 3251  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1st 4137  df-2nd 4138  df-1o 4191  df-oadd 4193  df-omul 4194  df-er 4319  df-ec 4321  df-qs 4324  df-ni 5065  df-pli 5066  df-mi 5067  df-lti 5068  df-plpq 5100  df-mpq 5101  df-enq 5102  df-nq