Theorem srglz 13336

Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)

Hypotheses

Ref	Expression
srgz.b	|- B = ( Base ` R )
srgz.t	|- .x. = ( .r ` R )
srgz.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
srglz	|- ( ( R e. SRing /\ X e. B ) -> ( .0. .x. X ) = .0. )

Proof of Theorem srglz

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		srgz.b	. . . . . . 7 |- B = ( Base ` R )
2		eqid 2189	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2189	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
4		srgz.t	. . . . . . 7 |- .x. = ( .r ` R )
5		srgz.z	. . . . . . 7 |- .0. = ( 0g ` R )
6	1, 2, 3, 4, 5	issrg 13316	. . . . . 6 |- ( R e. SRing <-> ( R e. CMnd /\ (mulGrp ` R ) e. Mnd /\ A. x e. B ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) ) )
7	6	simp3bi 1016	. . . . 5 |- ( R e. SRing -> A. x e. B ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) )
8	7	r19.21bi 2578	. . . 4 |- ( ( R e. SRing /\ x e. B ) -> ( A. y e. B A. z e. B ( ( x .x. ( y ( +g ` R ) z ) ) = ( ( x .x. y ) ( +g ` R ) ( x .x. z ) ) /\ ( ( x ( +g ` R ) y ) .x. z ) = ( ( x .x. z ) ( +g ` R ) ( y .x. z ) ) ) /\ ( ( .0. .x. x ) = .0. /\ ( x .x. .0. ) = .0. ) ) )
9	8	simprld 530	. . 3 |- ( ( R e. SRing /\ x e. B ) -> ( .0. .x. x ) = .0. )
10	9	ralrimiva 2563	. 2 |- ( R e. SRing -> A. x e. B ( .0. .x. x ) = .0. )
11		oveq2 5903	. . . 4 |- ( x = X -> ( .0. .x. x ) = ( .0. .x. X ) )
12	11	eqeq1d 2198	. . 3 |- ( x = X -> ( ( .0. .x. x ) = .0. <-> ( .0. .x. X ) = .0. ) )
13	12	rspcv 2852	. 2 |- ( X e. B -> ( A. x e. B ( .0. .x. x ) = .0. -> ( .0. .x. X ) = .0. ) )
14	10, 13	mpan9 281	1 |- ( ( R e. SRing /\ X e. B ) -> ( .0. .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586   .rcmulr 12587   0gc0g 12758   Mndcmnd 12874  CMndccmn 13220  mulGrpcmgp 13271  SRingcsrg 13314

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5851  df-ov 5898  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-plusg 12599  df-mulr 12600  df-0g 12760  df-srg 13315

This theorem is referenced by:  srgmulgass  13340  srgrmhm  13345