Theorem 1idl 33825

Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

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

Assertion

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

Proof of Theorem 1idl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1idl.1	. . . . . 6 |- G = ( 1st ` R )
2		1idl.3	. . . . . 6 |- X = ran G
3	1, 2	idlss 33815	. . . . 5 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ X )
4	3	adantr 481	. . . 4 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ U e. I ) -> I C_ X )
5		1idl.2	. . . . . . . . 9 |- H = ( 2nd ` R )
6	1	rneqi 5352	. . . . . . . . . 10 |- ran G = ran ( 1st ` R )
7	2, 6	eqtri 2644	. . . . . . . . 9 |- X = ran ( 1st ` R )
8		1idl.4	. . . . . . . . 9 |- U = (GId ` H )
9	5, 7, 8	rngolidm 33736	. . . . . . . 8 |- ( ( R e. RingOps /\ x e. X ) -> ( U H x ) = x )
10	9	ad2ant2rl 785	. . . . . . 7 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ ( U e. I /\ x e. X ) ) -> ( U H x ) = x )
11	1, 5, 2	idlrmulcl 33820	. . . . . . 7 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ ( U e. I /\ x e. X ) ) -> ( U H x ) e. I )
12	10, 11	eqeltrrd 2702	. . . . . 6 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ ( U e. I /\ x e. X ) ) -> x e. I )
13	12	expr 643	. . . . 5 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ U e. I ) -> ( x e. X -> x e. I ) )
14	13	ssrdv 3609	. . . 4 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ U e. I ) -> X C_ I )
15	4, 14	eqssd 3620	. . 3 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ U e. I ) -> I = X )
16	15	ex 450	. 2 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( U e. I -> I = X ) )
17	7, 5, 8	rngo1cl 33738	. . . 4 |- ( R e. RingOps -> U e. X )
18	17	adantr 481	. . 3 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> U e. X )
19		eleq2 2690	. . 3 |- ( I = X -> ( U e. I <-> U e. X ) )
20	18, 19	syl5ibrcom 237	. 2 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( I = X -> U e. I ) )
21	16, 20	impbid 202	1 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( U e. I <-> I = X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   RingOpscrngo 33693   Idlcidl 33806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-idl 33809

This theorem is referenced by:  0rngo  33826  divrngidl  33827  maxidln1  33843