Theorem 1idsr 6359

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 6319	. 2 |- R. = ((P. X. P.)/. ~R )
2		opreq1 4889	. . 3 |- ([<.x, y>.] ~R = A -> ([<.x, y>.] ~R .R 1R) = (A .R 1R))
3		id 73	. . 3 |- ([<.x, y>.] ~R = A -> [<.x, y>.] ~R = A)
4	2, 3	eqeq12d 1899	. 2 |- ([<.x, y>.] ~R = A -> (([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R <-> (A .R 1R) = A))
5		1pr 6269	. . . . . 6 |- 1P e. P.
6		addclpr 6272	. . . . . 6 |- ((1P e. P. /\ 1P e. P.) -> (1P +P. 1P) e. P.)
7	5, 5, 6	mp2an 761	. . . . 5 |- (1P +P. 1P) e. P.
8		mulsrpr 6337	. . . . 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 )
9	7, 5, 8	mpanr12 778	. . . 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 )
10		1idpr 6285	. . . . . . . . 9 |- (x e. P. -> (x .P. 1P) = x)
11	10	opreq1d 4897	. . . . . . . 8 |- (x e. P. -> ((x .P. 1P) +P. (x .P. 1P)) = (x +P. (x .P. 1P)))
12	5	elisseti 2301	. . . . . . . . 9 |- 1P e. _V
13	12, 12	distrpr 6284	. . . . . . . 8 |- (x .P. (1P +P. 1P)) = ((x .P. 1P) +P. (x .P. 1P))
14	11, 13	syl5req 1941	. . . . . . 7 |- (x e. P. -> (x +P. (x .P. 1P)) = (x .P. (1P +P. 1P)))
15		1idpr 6285	. . . . . . . . 9 |- (y e. P. -> (y .P. 1P) = y)
16	15	opreq1d 4897	. . . . . . . 8 |- (y e. P. -> ((y .P. 1P) +P. (y .P. 1P)) = (y +P. (y .P. 1P)))
17	12, 12	distrpr 6284	. . . . . . . 8 |- (y .P. (1P +P. 1P)) = ((y .P. 1P) +P. (y .P. 1P))
18	16, 17	syl5eq 1940	. . . . . . 7 |- (y e. P. -> (y .P. (1P +P. 1P)) = (y +P. (y .P. 1P)))
19	14, 18	opreqan12d 4902	. . . . . 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))))
20		oprex 4907	. . . . . . 7 |- (x .P. 1P) e. _V
21		oprex 4907	. . . . . . 7 |- (y .P. (1P +P. 1P)) e. _V
22	20, 21	addasspr 6276	. . . . . 6 |- ((x +P. (x .P. 1P)) +P. (y .P. (1P +P. 1P))) = (x +P. ((x .P. 1P) +P. (y .P. (1P +P. 1P))))
23		oprex 4907	. . . . . . 7 |- (x .P. (1P +P. 1P)) e. _V
24		visset 2295	. . . . . . 7 |- y e. _V
25		oprex 4907	. . . . . . 7 |- (y .P. 1P) e. _V
26		visset 2295	. . . . . . . 8 |- z e. _V
27		visset 2295	. . . . . . . 8 |- w e. _V
28	26, 27	addcompr 6275	. . . . . . 7 |- (z +P. w) = (w +P. z)
29		visset 2295	. . . . . . . 8 |- v e. _V
30	27, 29	addasspr 6276	. . . . . . 7 |- ((z +P. w) +P. v) = (z +P. (w +P. v))
31	23, 24, 25, 28, 30	caopr12 4994	. . . . . 6 |- ((x .P. (1P +P. 1P)) +P. (y +P. (y .P. 1P))) = (y +P. ((x .P. (1P +P. 1P)) +P. (y .P. 1P)))
32	19, 22, 31	3eqtr3g 1952	. . . . 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))))
33		addclpr 6272	. . . . . . . . 9 |- (((x .P. (1P +P. 1P)) e. P. /\ (y .P. 1P) e. P.) -> ((x .P. (1P +P. 1P)) +P. (y .P. 1P)) e. P.)
34		mulclpr 6274	. . . . . . . . . 10 |- ((x e. P. /\ (1P +P. 1P) e. P.) -> (x .P. (1P +P. 1P)) e. P.)
35	7, 34	mpan2 760	. . . . . . . . 9 |- (x e. P. -> (x .P. (1P +P. 1P)) e. P.)
36		mulclpr 6274	. . . . . . . . . 10 |- ((y e. P. /\ 1P e. P.) -> (y .P. 1P) e. P.)
37	5, 36	mpan2 760	. . . . . . . . 9 |- (y e. P. -> (y .P. 1P) e. P.)
38	33, 35, 37	syl2an 503	. . . . . . . 8 |- ((x e. P. /\ y e. P.) -> ((x .P. (1P +P. 1P)) +P. (y .P. 1P)) e. P.)
39		addclpr 6272	. . . . . . . . 9 |- (((x .P. 1P) e. P. /\ (y .P. (1P +P. 1P)) e. P.) -> ((x .P. 1P) +P. (y .P. (1P +P. 1P))) e. P.)
40		mulclpr 6274	. . . . . . . . . 10 |- ((x e. P. /\ 1P e. P.) -> (x .P. 1P) e. P.)
41	5, 40	mpan2 760	. . . . . . . . 9 |- (x e. P. -> (x .P. 1P) e. P.)
42		mulclpr 6274	. . . . . . . . . 10 |- ((y e. P. /\ (1P +P. 1P) e. P.) -> (y .P. (1P +P. 1P)) e. P.)
43	7, 42	mpan2 760	. . . . . . . . 9 |- (y e. P. -> (y .P. (1P +P. 1P)) e. P.)
44	39, 41, 43	syl2an 503	. . . . . . . 8 |- ((x e. P. /\ y e. P.) -> ((x .P. 1P) +P. (y .P. (1P +P. 1P))) e. P.)
45	38, 44	anim12i 360	. . . . . . 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.))
46		enreceq 6329	. . . . . . 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)))))
47	45, 46	syldan 516	. . . . . 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)))))
48	47	anidms 480	. . . . 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)))))
49	32, 48	mpbird 213	. . . 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 )
50	9, 49	eqtr4d 1928	. . 3 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R ) = [<.x, y>.] ~R )
51		df-1r 6324	. . . 4 |- 1R = [<.(1P +P. 1P), 1P>.] ~R
52	51	opreq2i 4893	. . 3 |- ([<.x, y>.] ~R .R 1R) = ([<.x, y>.] ~R .R [<.(1P +P. 1P), 1P>.] ~R )
53	50, 52	syl5eq 1940	. 2 |- ((x e. P. /\ y e. P.) -> ([<.x, y>.] ~R .R 1R) = [<.x, y>.] ~R )
54	1, 4, 53	ecoptocl 5362	1 |- (A e. R. -> (A .R 1R) = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  <.cop 3046  (class class class)co 4884  [cec 5316  P.cnp 6137  1Pc1p 6138   +P. cpp 6139   .P. cmp 6140   ~R cer 6144  R.cnr 6145  1Rc1r 6147   .R cmr 6150

This theorem is referenced by:  pn0sr 6362  sqgt0sr 6367  ax1id 6435  axi2m1 6438  axcnre 6439

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-mpr 6317  df-enr 6318  df-nr 6319  df-mr 6321  df-1r 6324

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