Theorem rrgeq0i 13796

Description: Property of a left-regular element. (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
rrgeq0i	|- ( ( X e. E /\ Y e. B ) -> ( ( X .x. Y ) = .0. -> Y = .0. ) )

Proof of Theorem rrgeq0i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. . . 4 |- E = (RLReg ` R )
2		rrgval.b	. . . 4 |- B = ( Base ` R )
3		rrgval.t	. . . 4 |- .x. = ( .r ` R )
4		rrgval.z	. . . 4 |- .0. = ( 0g ` R )
5	1, 2, 3, 4	isrrg 13795	. . 3 |- ( X e. E <-> ( X e. B /\ A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) ) )
6	5	simprbi 275	. 2 |- ( X e. E -> A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) )
7		oveq2 5930	. . . . 5 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
8	7	eqeq1d 2205	. . . 4 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
9		eqeq1 2203	. . . 4 |- ( y = Y -> ( y = .0. <-> Y = .0. ) )
10	8, 9	imbi12d 234	. . 3 |- ( y = Y -> ( ( ( X .x. y ) = .0. -> y = .0. ) <-> ( ( X .x. Y ) = .0. -> Y = .0. ) ) )
11	10	rspcv 2864	. 2 |- ( Y e. B -> ( A. y e. B ( ( X .x. y ) = .0. -> y = .0. ) -> ( ( X .x. Y ) = .0. -> Y = .0. ) ) )
12	6, 11	mpan9 281	1 |- ( ( X e. E /\ Y e. B ) -> ( ( X .x. Y ) = .0. -> Y = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12654   .rcmulr 12732   0gc0g 12903  RLRegcrlreg 13787

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7968  ax-resscn 7969  ax-1re 7971  ax-addrcl 7974

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8988  df-ndx 12657  df-slot 12658  df-base 12660  df-rlreg 13790

This theorem is referenced by:  rrgeq0  13797  znrrg  14192