Theorem srgrz 14395

Description: The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

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

Assertion

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

Proof of Theorem srgrz

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 2238	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2238	. . . . . . 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 14376	. . . . . 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 1045	. . . . 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 2638	. . . 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	simprrd 538	. . 3 |- ( ( R e. SRing /\ x e. B ) -> ( x .x. .0. ) = .0. )
10	9	ralrimiva 2623	. 2 |- ( R e. SRing -> A. x e. B ( x .x. .0. ) = .0. )
11		oveq1 6095	. . . 4 |- ( x = X -> ( x .x. .0. ) = ( X .x. .0. ) )
12	11	eqeq1d 2247	. . 3 |- ( x = X -> ( ( x .x. .0. ) = .0. <-> ( X .x. .0. ) = .0. ) )
13	12	rspcv 2925	. 2 |- ( X e. B -> ( A. x e. B ( x .x. .0. ) = .0. -> ( X .x. .0. ) = .0. ) )
14	10, 13	mpan9 281	1 |- ( ( R e. SRing /\ X e. B ) -> ( X .x. .0. ) = .0. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   .rcmulr 13491   0gc0g 13669   Mndcmnd 13788  CMndccmn 14177  mulGrpcmgp 14324  SRingcsrg 14374

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof 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-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-riota 6041  df-ov 6091  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-mulr 13504  df-0g 13671  df-srg 14375

This theorem is used by:  srgisid  14397  srglmhm  14404