Theorem rrgval 14493

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 variables r z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. . . 4 |- E = (RLReg ` R )
2	1	rrgmex 14492	. . 3 |- ( z e. E -> R e. _V )
3		elrabi 2973	. . . 4 |- ( z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } -> z e. B )
4		rrgval.b	. . . . 5 |- B = ( Base ` R )
5	4	basmex 13356	. . . 4 |- ( z e. B -> R e. _V )
6	3, 5	syl 14	. . 3 |- ( z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } -> R e. _V )
7		df-rlreg 14489	. . . . . 6 |- RLReg = ( r e. _V |-> { x e. ( Base ` r ) | A. y e. ( Base ` r ) ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) } )
8		fveq2 5675	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9	8, 4	eqtr4di 2285	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
10		fveq2 5675	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
11		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12	10, 11	eqtr4di 2285	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = .x. )
13	12	oveqd 6075	. . . . . . . . . 10 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
14		fveq2 5675	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
15		rrgval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
16	14, 15	eqtr4di 2285	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
17	13, 16	eqeq12d 2249	. . . . . . . . 9 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
18	16	eqeq2d 2246	. . . . . . . . 9 |- ( r = R -> ( y = ( 0g ` r ) <-> y = .0. ) )
19	17, 18	imbi12d 234	. . . . . . . 8 |- ( r = R -> ( ( ( x ( .r ` r ) y ) = ( 0g ` r ) -> y = ( 0g ` r ) ) <-> ( ( x .x. y ) = .0. -> y = .0. ) ) )
20	9, 19	raleqbidv 2759	. . . . . . 7 |- ( 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. ) ) )
21	9, 20	rabeqbidv 2810	. . . . . 6 |- ( 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. ) } )
22		id 19	. . . . . 6 |- ( R e. _V -> R e. _V )
23		basfn 13355	. . . . . . . . 9 |- Base Fn _V
24		funfvex 5692	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5463	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
26	23, 25	mpan 424	. . . . . . . 8 |- ( R e. _V -> ( Base ` R ) e. _V )
27	4, 26	eqeltrid 2321	. . . . . . 7 |- ( R e. _V -> B e. _V )
28		rabexg 4260	. . . . . . 7 |- ( B e. _V -> { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } e. _V )
29	27, 28	syl 14	. . . . . 6 |- ( R e. _V -> { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } e. _V )
30	7, 21, 22, 29	fvmptd3 5776	. . . . 5 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
31	1, 30	eqtrid 2279	. . . 4 |- ( R e. _V -> E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
32	31	eleq2d 2304	. . 3 |- ( R e. _V -> ( z e. E <-> z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } ) )
33	2, 6, 32	pm5.21nii 712	. 2 |- ( z e. E <-> z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
34	33	eqriv 2231	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   _Vcvv 2815   Fn wfn 5352   ` 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:  isrrg  14494  rrgeq0  14496  rrgss  14498