Theorem rrgeq0i 14555

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 14554	. . 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 6087	. . . . 5 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
8	7	eqeq1d 2247	. . . 4 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
9		eqeq1 2245	. . . 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 2925	. 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 1402   e. wcel 2209   A.wral 2528   ` 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:  rrgeq0  14556  znrrg  14978