Theorem ringidval 17149

Description: The value of the identity element of a group.

Hypotheses

Ref	Expression
ringidval.b	|- B = (base` R)
ringidval.t	|- T = (.r` R)
ringidval.u	|- U = (1rNEW` R)

Assertion

Ref	Expression
ringidval	|- (R e. A -> U = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))

Distinct variable groups:   x,B   x,u,R

Proof of Theorem ringidval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (R e. A -> R e. _V)
2		fveq2 4681	. . . . . . . 8 |- (r = R -> (base` r) = (base` R))
3		ringidval.b	. . . . . . . 8 |- B = (base` R)
4	2, 3	syl6eqr 1946	. . . . . . 7 |- (r = R -> (base` r) = B)
5	4	eleq2d 1964	. . . . . 6 |- (r = R -> (u e. (base` r) <-> u e. B))
6		fveq2 4681	. . . . . . . . . . 11 |- (r = R -> (.r` r) = (.r` R))
7		ringidval.t	. . . . . . . . . . 11 |- T = (.r` R)
8	6, 7	syl6eqr 1946	. . . . . . . . . 10 |- (r = R -> (.r` r) = T)
9	8	opreqd 4899	. . . . . . . . 9 |- (r = R -> (u(.r` r)x) = (uTx))
10	9	eqeq1d 1892	. . . . . . . 8 |- (r = R -> ((u(.r` r)x) = x <-> (uTx) = x))
11	8	opreqd 4899	. . . . . . . . 9 |- (r = R -> (x(.r` r)u) = (xTu))
12	11	eqeq1d 1892	. . . . . . . 8 |- (r = R -> ((x(.r` r)u) = x <-> (xTu) = x))
13	10, 12	anbi12d 690	. . . . . . 7 |- (r = R -> (((u(.r` r)x) = x /\ (x(.r` r)u) = x) <-> ((uTx) = x /\ (xTu) = x)))
14	4, 13	raleqbidv 2274	. . . . . 6 |- (r = R -> (A.x e. (base` r)((u(.r` r)x) = x /\ (x(.r` r)u) = x) <-> A.x e. B ((uTx) = x /\ (xTu) = x)))
15	5, 14	anbi12d 690	. . . . 5 |- (r = R -> ((u e. (base` r) /\ A.x e. (base` r)((u(.r` r)x) = x /\ (x(.r` r)u) = x)) <-> (u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))
16	15	iotabidv 5102	. . . 4 |- (r = R -> (iotau(u e. (base` r) /\ A.x e. (base` r)((u(.r` r)x) = x /\ (x(.r` r)u) = x))) = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))
17		df-ur 17095	. . . 4 |- 1rNEW = (r e. _V |-> (iotau(u e. (base` r) /\ A.x e. (base` r)((u(.r` r)x) = x /\ (x(.r` r)u) = x))))
18		iotaex 5099	. . . 4 |- (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))) e. _V
19	16, 17, 18	fvmpt 5015	. . 3 |- (R e. _V -> (1rNEW` R) = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))
20		ringidval.u	. . 3 |- U = (1rNEW` R)
21	19, 20	syl5eq 1940	. 2 |- (R e. _V -> U = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))
22	1, 21	syl 12	1 |- (R e. A -> U = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  ` cfv 3998  (class class class)co 4884  iotacio 5087  basecbs 16758  ._rcmulr 17085  1_rNEWcur 17087

This theorem is referenced by:  ringidcl 17150  ringidmlemNEW 17153

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-opr 4886  df-mpt 5006  df-iota 5089  df-ur 17095

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