Theorem rrgval 14282

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 14281	. . 3 |- ( z e. E -> R e. _V )
3		elrabi 2959	. . . 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 13147	. . . 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 14278	. . . . . 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 5639	. . . . . . . 8 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9	8, 4	eqtr4di 2282	. . . . . . 7 |- ( r = R -> ( Base ` r ) = B )
10		fveq2 5639	. . . . . . . . . . . 12 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
11		rrgval.t	. . . . . . . . . . . 12 |- .x. = ( .r ` R )
12	10, 11	eqtr4di 2282	. . . . . . . . . . 11 |- ( r = R -> ( .r ` r ) = .x. )
13	12	oveqd 6035	. . . . . . . . . 10 |- ( r = R -> ( x ( .r ` r ) y ) = ( x .x. y ) )
14		fveq2 5639	. . . . . . . . . . 11 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
15		rrgval.z	. . . . . . . . . . 11 |- .0. = ( 0g ` R )
16	14, 15	eqtr4di 2282	. . . . . . . . . 10 |- ( r = R -> ( 0g ` r ) = .0. )
17	13, 16	eqeq12d 2246	. . . . . . . . 9 |- ( r = R -> ( ( x ( .r ` r ) y ) = ( 0g ` r ) <-> ( x .x. y ) = .0. ) )
18	16	eqeq2d 2243	. . . . . . . . 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 2746	. . . . . . 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 2797	. . . . . 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 13146	. . . . . . . . 9 |- Base Fn _V
24		funfvex 5656	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
25	24	funfni 5432	. . . . . . . . 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 2318	. . . . . . 7 |- ( R e. _V -> B e. _V )
28		rabexg 4233	. . . . . . 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 5740	. . . . 5 |- ( R e. _V -> (RLReg ` R ) = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
31	1, 30	eqtrid 2276	. . . 4 |- ( R e. _V -> E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
32	31	eleq2d 2301	. . 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 711	. 2 |- ( z e. E <-> z e. { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) } )
34	33	eqriv 2228	1 |- E = { x e. B | A. y e. B ( ( x .x. y ) = .0. -> y = .0. ) }

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   _Vcvv 2802   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   Basecbs 13087   .rcmulr 13166   0gc0g 13344  RLRegcrlreg 14275

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-inn 9144  df-ndx 13090  df-slot 13091  df-base 13093  df-rlreg 14278

This theorem is referenced by:  isrrg  14283  rrgeq0  14285  rrgss  14286