Theorem rrgval 13828

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 13827	. . 3 |- ( z e. E -> R e. _V )
3		elrabi 2917	. . . 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 12747	. . . 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 13824	. . . . . 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 5559	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9	8, 4	eqtr4di 2247	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
10		fveq2 5559	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
11		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12	10, 11	eqtr4di 2247	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = .x. )
13	12	oveqd 5940	. . . . . . . . . 10 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
14		fveq2 5559	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
15		rrgval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
16	14, 15	eqtr4di 2247	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
17	13, 16	eqeq12d 2211	. . . . . . . . 9 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
18	16	eqeq2d 2208	. . . . . . . . 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 2709	. . . . . . 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 2758	. . . . . 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 12746	. . . . . . . . 9 |- Base Fn _V
24		funfvex 5576	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5359	. . . . . . . . 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 2283	. . . . . . 7 |- ( R e. _V -> B e. _V )
28		rabexg 4177	. . . . . . 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 5656	. . . . 5 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
31	1, 30	eqtrid 2241	. . . 4 |- ( R e. _V -> E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
32	31	eleq2d 2266	. . 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 705	. 2 |- ( z e. E <-> z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
34	33	eqriv 2193	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763   Fn wfn 5254   ` cfv 5259  (class class class)co 5923   Basecbs 12688   .rcmulr 12766   0gc0g 12937  RLRegcrlreg 13821

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7972  ax-resscn 7973  ax-1re 7975  ax-addrcl 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5926  df-inn 8993  df-ndx 12691  df-slot 12692  df-base 12694  df-rlreg 13824

This theorem is referenced by:  isrrg  13829  rrgeq0  13831  rrgss  13832