Theorem isrrg 14494

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 6065	. . . . 5 |- ( x = X -> ( x .x. y ) = ( X .x. y ) )
2	1	eqeq1d 2243	. . . 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 2544	. 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 14493	. 2 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
10	4, 9	elrab2 2979	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 1398   e. wcel 2205   A.wral 2522   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375   0gc0g 13553  RLRegcrlreg 14486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-rlreg 14489

This theorem is referenced by:  rrgeq0i  14495  rrgsupp  14497  unitrrg  14499