Theorem ringidmlemNEW 17153

Description: Lemma for ringlidmNEW 17154 and ringridmNEW 17155.

Hypotheses

Ref	Expression
ringidm.1NEW	|- B = (base` R)
ringidm.3NEW	|- T = (.r` R)
ringidm.4NEW	|- U = (1rNEW` R)

Assertion

Ref	Expression
ringidmlemNEW	|- ((R e. RingNEW /\ X e. B) -> ((UTX) = X /\ (XTU) = X))

Proof of Theorem ringidmlemNEW

Step	Hyp	Ref	Expression
1		ringidm.1NEW	. . . . 5 |- B = (base` R)
2		ringidm.3NEW	. . . . 5 |- T = (.r` R)
3		ringidm.4NEW	. . . . 5 |- U = (1rNEW` R)
4	1, 2, 3	ringidval 17149	. . . 4 |- (R e. RingNEW -> U = (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))))
5	4	eqcomd 1889	. . 3 |- (R e. RingNEW -> (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))) = U)
6	1, 3	ringidcl 17150	. . . 4 |- (R e. RingNEW -> U e. B)
7	1, 2	ringideuNEW 17146	. . . 4 |- (R e. RingNEW -> E!u e. B A.x e. B ((uTx) = x /\ (xTu) = x))
8		opreq1 4889	. . . . . . . 8 |- (u = U -> (uTx) = (UTx))
9	8	eqeq1d 1892	. . . . . . 7 |- (u = U -> ((uTx) = x <-> (UTx) = x))
10		opreq2 4890	. . . . . . . 8 |- (u = U -> (xTu) = (xTU))
11	10	eqeq1d 1892	. . . . . . 7 |- (u = U -> ((xTu) = x <-> (xTU) = x))
12	9, 11	anbi12d 690	. . . . . 6 |- (u = U -> (((uTx) = x /\ (xTu) = x) <-> ((UTx) = x /\ (xTU) = x)))
13	12	ralbidv 2123	. . . . 5 |- (u = U -> (A.x e. B ((uTx) = x /\ (xTu) = x) <-> A.x e. B ((UTx) = x /\ (xTU) = x)))
14	13	reiota2 5110	. . . 4 |- ((U e. B /\ E!u e. B A.x e. B ((uTx) = x /\ (xTu) = x)) -> (A.x e. B ((UTx) = x /\ (xTU) = x) <-> (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))) = U))
15	6, 7, 14	syl11anc 524	. . 3 |- (R e. RingNEW -> (A.x e. B ((UTx) = x /\ (xTU) = x) <-> (iotau(u e. B /\ A.x e. B ((uTx) = x /\ (xTu) = x))) = U))
16	5, 15	mpbird 213	. 2 |- (R e. RingNEW -> A.x e. B ((UTx) = x /\ (xTU) = x))
17		opreq2 4890	. . . . 5 |- (x = X -> (UTx) = (UTX))
18		id 73	. . . . 5 |- (x = X -> x = X)
19	17, 18	eqeq12d 1899	. . . 4 |- (x = X -> ((UTx) = x <-> (UTX) = X))
20		opreq1 4889	. . . . 5 |- (x = X -> (xTU) = (XTU))
21	20, 18	eqeq12d 1899	. . . 4 |- (x = X -> ((xTU) = x <-> (XTU) = X))
22	19, 21	anbi12d 690	. . 3 |- (x = X -> (((UTx) = x /\ (xTU) = x) <-> ((UTX) = X /\ (XTU) = X)))
23	22	rcla4v 2376	. 2 |- (X e. B -> (A.x e. B ((UTx) = x /\ (xTU) = x) -> ((UTX) = X /\ (XTU) = X)))
24	16, 23	mpan9 521	1 |- ((R e. RingNEW /\ X e. B) -> ((UTX) = X /\ (XTU) = X))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E!wreu 2107  ` cfv 3998  (class class class)co 4884  iotacio 5087  basecbs 16758  ._rcmulr 17085  RingNEWcrg 17086  1_rNEWcur 17087

This theorem is referenced by:  ringlidmNEW 17154  ringridmNEW 17155

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  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-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-fn 4009  df-fv 4014  df-opr 4886  df-mpt 5006  df-iota 5089  df-struct 16708  df-ringNEW 17094  df-ur 17095

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