Theorem rrgeq0i 14096

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 14095	. . 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 5964	. . . . 5 |- ( y = Y -> ( X .x. y ) = ( X .x. Y ) )
8	7	eqeq1d 2215	. . . 4 |- ( y = Y -> ( ( X .x. y ) = .0. <-> ( X .x. Y ) = .0. ) )
9		eqeq1 2213	. . . 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 2877	. 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 1373   e. wcel 2177   A.wral 2485   ` cfv 5279  (class class class)co 5956   Basecbs 12902   .rcmulr 12980   0gc0g 13158  RLRegcrlreg 14087

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-cnex 8031  ax-resscn 8032  ax-1re 8034  ax-addrcl 8037

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-iota 5240  df-fun 5281  df-fn 5282  df-fv 5287  df-ov 5959  df-inn 9052  df-ndx 12905  df-slot 12906  df-base 12908  df-rlreg 14090

This theorem is referenced by:  rrgeq0  14097  znrrg  14492