Theorem isrrg 14283

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 6025	. . . . 5 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
2	1	eqeq1d 2240	. . . 4 |- ( x = X -> ( ( x .x. y ) = .0. <-> ( X .x. y ) = .0. ) )
3	2	imbi1d 231	. . 3 |- ( x = X -> ( ( ( x .x. y ) = .0. -> y = .0. ) <-> ( ( X .x. y ) = .0. -> y = .0. ) ) )
4	3	ralbidv 2532	. 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 14282	. 2 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
10	4, 9	elrab2 2965	1 |- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6018   Basecbs 13087   .rcmulr 13166   0gc0g 13344  RLRegcrlreg 14275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-inn 9144  df-ndx 13090  df-slot 13091  df-base 13093  df-rlreg 14278

This theorem is referenced by:  rrgeq0i  14284  unitrrg  14287