Theorem isrrg 14339

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 6035	. . . . 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 2533	. 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 14338	. 2 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
10	4, 9	elrab2 2966	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 2202   A.wral 2511   ` cfv 5333  (class class class)co 6028   Basecbs 13143   .rcmulr 13222   0gc0g 13400  RLRegcrlreg 14331

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-inn 9187  df-ndx 13146  df-slot 13147  df-base 13149  df-rlreg 14334

This theorem is referenced by:  rrgeq0i  14340  unitrrg  14343