Theorem rrgval 14553

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 14552	. . 3 |- ( z e. E -> R e. _V )
3		elrabi 2979	. . . 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 13395	. . . 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 14549	. . . . . 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 5693	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9	8, 4	eqtr4di 2289	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
10		fveq2 5693	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
11		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12	10, 11	eqtr4di 2289	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = .x. )
13	12	oveqd 6096	. . . . . . . . . 10 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
14		fveq2 5693	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
15		rrgval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
16	14, 15	eqtr4di 2289	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
17	13, 16	eqeq12d 2253	. . . . . . . . 9 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
18	16	eqeq2d 2250	. . . . . . . . 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 2765	. . . . . . 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 2816	. . . . . 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 13394	. . . . . . . . 9 |- Base Fn _V
24		funfvex 5710	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5481	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
26	23, 25	mpan 428	. . . . . . . 8 |- ( R e. _V -> ( Base ` R ) e. _V )
27	4, 26	eqeltrid 2325	. . . . . . 7 |- ( R e. _V -> B e. _V )
28		rabexg 4277	. . . . . . 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 5796	. . . . 5 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
31	1, 30	eqtrid 2283	. . . 4 |- ( R e. _V -> E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
32	31	eleq2d 2308	. . 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 716	. 2 |- ( z e. E <-> z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
34	33	eqriv 2235	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   Basecbs 13335   .rcmulr 13415   0gc0g 13593  RLRegcrlreg 14546

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-rlreg 14549

This theorem is referenced by:  isrrg  14554  rrgeq0  14556  rrgss  14558