Theorem rrgval 19287

Description: Value of the set or left-regular elements in a ring. (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
rrgval	|- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

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

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

Proof of Theorem rrgval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. 2 |- E = (RLReg ` R )
2		fveq2 6191	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
3		rrgval.b	. . . . . 6 |- B = ( Base ` R )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( r = R -> ( Base ` r ) = B )
5		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
6		rrgval.t	. . . . . . . . . 10 |- .x. = ( .r ` R )
7	5, 6	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( .r ` r ) = .x. )
8	7	oveqd 6667	. . . . . . . 8 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
9		fveq2 6191	. . . . . . . . 9 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
10		rrgval.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
11	9, 10	syl6eqr 2674	. . . . . . . 8 |- ( r = R -> ( 0g ` r ) = .0. )
12	8, 11	eqeq12d 2637	. . . . . . 7 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
13	11	eqeq2d 2632	. . . . . . 7 |- ( r = R -> ( y = ( 0g ` r ) <-> y = .0. ) )
14	12, 13	imbi12d 334	. . . . . 6 |- ( r = R -> ( ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) <-> ( ( x .x. y ) = .0. -> y = .0. ) ) )
15	4, 14	raleqbidv 3152	. . . . 5 |- ( r = R -> ( A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) <-> A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) ) )
16	4, 15	rabeqbidv 3195	. . . 4 |- ( r = R -> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
17		df-rlreg 19283	. . . 4 |- RLReg = ( r e. _V |-> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } )
18		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
19	3, 18	eqeltri 2697	. . . . 5 |- B e. _V
20	19	rabex 4813	. . . 4 |- { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } e. _V
21	16, 17, 20	fvmpt 6282	. . 3 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
22		fvprc 6185	. . . 4 |- ( -. R e. _V -> (RLReg ` R ) = (/) )
23		fvprc 6185	. . . . . . 7 |- ( -. R e. _V -> ( Base ` R ) = (/) )
24	3, 23	syl5eq 2668	. . . . . 6 |- ( -. R e. _V -> B = (/) )
25		rabeq 3192	. . . . . 6 |- ( B = (/) -> { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } = { x e. (/) | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
26	24, 25	syl 17	. . . . 5 |- ( -. R e. _V -> { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } = { x e. (/) | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
27		rab0 3955	. . . . 5 |- { x e. (/) | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } = (/)
28	26, 27	syl6eq 2672	. . . 4 |- ( -. R e. _V -> { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } = (/) )
29	22, 28	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
30	21, 29	pm2.61i 176	. 2 |- (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }
31	1, 30	eqtri 2644	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   (/)c0 3915   ` 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:  isrrg  19288  rrgeq0  19290  rrgss  19292