Theorem idladdcl 33818

Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypothesis

Ref	Expression
idladdcl.1	|- G = ( 1st ` R )

Assertion

Ref	Expression
idladdcl	|- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ ( A e. I /\ B e. I ) ) -> ( A G B ) e. I )

Proof of Theorem idladdcl

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idladdcl.1	. . . . . 6 |- G = ( 1st ` R )
2		eqid 2622	. . . . . 6 |- ( 2nd ` R ) = ( 2nd ` R )
3		eqid 2622	. . . . . 6 |- ran G = ran G
4		eqid 2622	. . . . . 6 |- (GId ` G ) = (GId ` G )
5	1, 2, 3, 4	isidl 33813	. . . . 5 |- ( R e. RingOps -> ( I e. ( Idl ` R ) <-> ( I C_ ran G /\ (GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) ) )
6	5	biimpa 501	. . . 4 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( I C_ ran G /\ (GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) )
7	6	simp3d 1075	. . 3 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) )
8		simpl 473	. . . 4 |- ( ( A. y e. I ( x G y ) e. I /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) -> A. y e. I ( x G y ) e. I )
9	8	ralimi 2952	. . 3 |- ( A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. ran G ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) -> A. x e. I A. y e. I ( x G y ) e. I )
10	7, 9	syl 17	. 2 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> A. x e. I A. y e. I ( x G y ) e. I )
11		oveq1 6657	. . . 4 |- ( x = A -> ( x G y ) = ( A G y ) )
12	11	eleq1d 2686	. . 3 |- ( x = A -> ( ( x G y ) e. I <-> ( A G y ) e. I ) )
13		oveq2 6658	. . . 4 |- ( y = B -> ( A G y ) = ( A G B ) )
14	13	eleq1d 2686	. . 3 |- ( y = B -> ( ( A G y ) e. I <-> ( A G B ) e. I ) )
15	12, 14	rspc2v 3322	. 2 |- ( ( A e. I /\ B e. I ) -> ( A. x e. I A. y e. I ( x G y ) e. I -> ( A G B ) e. I ) )
16	10, 15	mpan9 486	1 |- ( ( ( R e. RingOps /\ I e. ( Idl ` R ) ) /\ ( A e. I /\ B e. I ) ) -> ( A G B ) e. I )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-idl 33809

This theorem is referenced by:  idlsubcl  33822  intidl  33828  unichnidl  33830