Theorem ringidmlem 10409

Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.)

Hypotheses

Ref	Expression
uridm.1	|- H = (2nd` R)
uridm.2	|- X = ran (1st` R)
uridm.3	|- U = (Id` H)

Assertion

Ref	Expression
ringidmlem	|- ((R e. Ring /\ A e. X) -> ((UHA) = A /\ (AHU) = A))

Proof of Theorem ringidmlem

Step	Hyp	Ref	Expression
1		uridm.1	. . . . 5 |- H = (2nd` R)
2	1	unmnd 10405	. . . 4 |- (R e. Ring -> H e. Mnd)
3		mndmgmid 10389	. . . 4 |- (H e. Mnd -> H e. (Magma i^i ExId ))
4		eqid 1884	. . . . . 6 |- ran H = ran H
5		uridm.3	. . . . . 6 |- U = (Id` H)
6	4, 5	cmpidelt 10376	. . . . 5 |- ((H e. (Magma i^i ExId ) /\ A e. ran H) -> ((UHA) = A /\ (AHU) = A))
7	6	ex 402	. . . 4 |- (H e. (Magma i^i ExId ) -> (A e. ran H -> ((UHA) = A /\ (AHU) = A)))
8	2, 3, 7	3syl 24	. . 3 |- (R e. Ring -> (A e. ran H -> ((UHA) = A /\ (AHU) = A)))
9		eqid 1884	. . . . 5 |- (1st` R) = (1st` R)
10	1, 9	rnplrnml 10404	. . . 4 |- (R e. Ring -> ran (1st` R) = ran H)
11		uridm.2	. . . . 5 |- X = ran (1st` R)
12		eqtr 1904	. . . . . 6 |- ((X = ran (1st` R) /\ ran (1st` R) = ran H) -> X = ran H)
13		simpl 346	. . . . . . . . 9 |- ((X = ran H /\ R e. Ring) -> X = ran H)
14	13	eleq2d 1964	. . . . . . . 8 |- ((X = ran H /\ R e. Ring) -> (A e. X <-> A e. ran H))
15	14	imbi1d 675	. . . . . . 7 |- ((X = ran H /\ R e. Ring) -> ((A e. X -> ((UHA) = A /\ (AHU) = A)) <-> (A e. ran H -> ((UHA) = A /\ (AHU) = A))))
16	15	ex 402	. . . . . 6 |- (X = ran H -> (R e. Ring -> ((A e. X -> ((UHA) = A /\ (AHU) = A)) <-> (A e. ran H -> ((UHA) = A /\ (AHU) = A)))))
17	12, 16	syl 12	. . . . 5 |- ((X = ran (1st` R) /\ ran (1st` R) = ran H) -> (R e. Ring -> ((A e. X -> ((UHA) = A /\ (AHU) = A)) <-> (A e. ran H -> ((UHA) = A /\ (AHU) = A)))))
18	11, 17	mpan 759	. . . 4 |- (ran (1st` R) = ran H -> (R e. Ring -> ((A e. X -> ((UHA) = A /\ (AHU) = A)) <-> (A e. ran H -> ((UHA) = A /\ (AHU) = A)))))
19	10, 18	mpcom 60	. . 3 |- (R e. Ring -> ((A e. X -> ((UHA) = A /\ (AHU) = A)) <-> (A e. ran H -> ((UHA) = A /\ (AHU) = A))))
20	8, 19	mpbird 213	. 2 |- (R e. Ring -> (A e. X -> ((UHA) = A /\ (AHU) = A)))
21	20	imp 377	1 |- ((R e. Ring /\ A e. X) -> ((UHA) = A /\ (AHU) = A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   i^i cin 2592  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463   ExId cexid 10361  Magmacmagm 10365  Mndcmnd 10384

This theorem is referenced by:  ringlidm 10410  ringridm 10411  uznzr 10416

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-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-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-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-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-abl 9408  df-ring 9464  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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