Theorem srglz 12961

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 2175	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2175	. . . . . . 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 12941	. . . . . 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 1014	. . . . 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 2563	. . . 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 2548	. 2 |- ( R e. SRing -> A. x e. B ( .0. .x. x ) = .0. )
11		oveq2 5873	. . . 4 |- ( x = X -> ( .0. .x. x ) = ( .0. .x. X ) )
12	11	eqeq1d 2184	. . 3 |- ( x = X -> ( ( .0. .x. x ) = .0. <-> ( .0. .x. X ) = .0. ) )
13	12	rspcv 2835	. 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 1353   e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12428   +g cplusg 12492   .rcmulr 12493   0gc0g 12626   Mndcmnd 12682  CMndccmn 12884  mulGrpcmgp 12925  SRingcsrg 12939

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-riota 5821  df-ov 5868  df-inn 8891  df-2 8949  df-3 8950  df-ndx 12431  df-slot 12432  df-base 12434  df-plusg 12505  df-mulr 12506  df-0g 12628  df-srg 12940

This theorem is referenced by:  srgmulgass  12965  srgrmhm  12970