Theorem 1idl 16174

Description: Two ways of expressing the unit ideal.

Hypotheses

Ref	Expression
1idl.1	|- G = (1st` R)
1idl.2	|- H = (2nd` R)
1idl.3	|- X = ran G
1idl.4	|- U = (Id` H)

Assertion

Ref	Expression
1idl	|- ((R e. Ring /\ I e. (Idl` R)) -> (U e. I <-> I = X))

Proof of Theorem 1idl

Step	Hyp	Ref	Expression
1		1idl.1	. . . . . 6 |- G = (1st` R)
2		1idl.3	. . . . . 6 |- X = ran G
3	1, 2	idlss 16164	. . . . 5 |- ((R e. Ring /\ I e. (Idl` R)) -> I C_ X)
4	3	adantr 425	. . . 4 |- (((R e. Ring /\ I e. (Idl` R)) /\ U e. I) -> I C_ X)
5		1idl.2	. . . . . . . . 9 |- H = (2nd` R)
6	1	rneqi 4187	. . . . . . . . . 10 |- ran G = ran (1st` R)
7	2, 6	eqtri 1908	. . . . . . . . 9 |- X = ran (1st` R)
8		1idl.4	. . . . . . . . 9 |- U = (Id` H)
9	5, 7, 8	ringlidm 10410	. . . . . . . 8 |- ((R e. Ring /\ x e. X) -> (UHx) = x)
10	9	ad2ant2rl 447	. . . . . . 7 |- (((R e. Ring /\ I e. (Idl` R)) /\ (U e. I /\ x e. X)) -> (UHx) = x)
11	1, 5, 2	idlrmulcl 16169	. . . . . . 7 |- (((R e. Ring /\ I e. (Idl` R)) /\ (U e. I /\ x e. X)) -> (UHx) e. I)
12	10, 11	eqeltrrd 1972	. . . . . 6 |- (((R e. Ring /\ I e. (Idl` R)) /\ (U e. I /\ x e. X)) -> x e. I)
13	12	expr 418	. . . . 5 |- (((R e. Ring /\ I e. (Idl` R)) /\ U e. I) -> (x e. X -> x e. I))
14	13	ssrdv 2622	. . . 4 |- (((R e. Ring /\ I e. (Idl` R)) /\ U e. I) -> X C_ I)
15	4, 14	eqssd 2633	. . 3 |- (((R e. Ring /\ I e. (Idl` R)) /\ U e. I) -> I = X)
16	15	ex 402	. 2 |- ((R e. Ring /\ I e. (Idl` R)) -> (U e. I -> I = X))
17		eleq2 1958	. . 3 |- (I = X -> (U e. I <-> U e. X))
18	7, 5, 8	ring1cl 10415	. . . 4 |- (R e. Ring -> U e. X)
19	18	adantr 425	. . 3 |- ((R e. Ring /\ I e. (Idl` R)) -> U e. X)
20	17, 19	syl5cbir 228	. 2 |- ((R e. Ring /\ I e. (Idl` R)) -> (I = X -> U e. I))
21	16, 20	impbid 574	1 |- ((R e. Ring /\ I e. (Idl` R)) -> (U e. I <-> I = X))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463  Idlcidl 16155

This theorem is referenced by:  0ring 16175  divrngidl 16176  maxidln1 16192

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  df-idl 16158

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