Theorem srgrz 13540

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 2196	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2196	. . . . . . 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 13521	. . . . . 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 2585	. . . 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 532	. . 3 |- ( ( R e. SRing /\ x e. B ) -> ( x .x. .0. ) = .0. )
10	9	ralrimiva 2570	. 2 |- ( R e. SRing -> A. x e. B ( x .x. .0. ) = .0. )
11		oveq1 5929	. . . 4 |- ( x = X -> ( x .x. .0. ) = ( X .x. .0. ) )
12	11	eqeq1d 2205	. . 3 |- ( x = X -> ( ( x .x. .0. ) = .0. <-> ( X .x. .0. ) = .0. ) )
13	12	rspcv 2864	. 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. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   0gc0g 12927   Mndcmnd 13057  CMndccmn 13414  mulGrpcmgp 13476  SRingcsrg 13519

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-0g 12929  df-srg 13520

This theorem is referenced by:  srgisid  13542  srglmhm  13549