Theorem rrgval 13742

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 13741	. . 3 |- ( z e. E -> R e. _V )
3		elrabi 2913	. . . 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 12667	. . . 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 13738	. . . . . 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 5546	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9	8, 4	eqtr4di 2244	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
10		fveq2 5546	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
11		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12	10, 11	eqtr4di 2244	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = .x. )
13	12	oveqd 5927	. . . . . . . . . 10 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
14		fveq2 5546	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
15		rrgval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
16	14, 15	eqtr4di 2244	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
17	13, 16	eqeq12d 2208	. . . . . . . . 9 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
18	16	eqeq2d 2205	. . . . . . . . 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 2706	. . . . . . 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 2755	. . . . . 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 12666	. . . . . . . . 9 |- Base Fn _V
24		funfvex 5563	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5346	. . . . . . . . 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 2280	. . . . . . 7 |- ( R e. _V -> B e. _V )
28		rabexg 4172	. . . . . . 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 5643	. . . . 5 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
31	1, 30	eqtrid 2238	. . . 4 |- ( R e. _V -> E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
32	31	eleq2d 2263	. . 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 2190	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   A.wral 2472   {crab 2476   _Vcvv 2760   Fn wfn 5241   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686   0gc0g 12857  RLRegcrlreg 13735

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-ov 5913  df-inn 8973  df-ndx 12611  df-slot 12612  df-base 12614  df-rlreg 13738

This theorem is referenced by:  isrrg  13743  rrgeq0  13745  rrgss  13746