Theorem isrrg 19288

Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgval.e	|- E = (RLReg ` R )
rrgval.b	|- B = ( Base ` R )
rrgval.t	|- .x. = ( .r ` R )
rrgval.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
isrrg	|- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )

Distinct variable groups:   y, B   y, R   y, X

Allowed substitution hints:   .x. ( y)   E( y)   .0. ( y)

Proof of Theorem isrrg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
2	1	eqeq1d 2624	. . . 4 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
3	2	imbi1d 331	. . 3 |- ( x = X -> ( ( ( x .x. y ) = .0. -> y = .0. ) <-> ( ( X .x. y ) = .0. -> y = .0. ) ) )
4	3	ralbidv 2986	. 2 |- ( x = X -> ( A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) <-> A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )
5		rrgval.e	. . 3 |- E = (RLReg ` R )
6		rrgval.b	. . 3 |- B = ( Base ` R )
7		rrgval.t	. . 3 |- .x. = ( .r ` R )
8		rrgval.z	. . 3 |- .0. = ( 0g ` R )
9	5, 6, 7, 8	rrgval 19287	. 2 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
10	4, 9	elrab2 3366	1 |- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100  RLRegcrlreg 19279

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

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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-rlreg 19283

This theorem is referenced by:  rrgeq0i  19289  unitrrg  19293  isdomn2  19299