Theorem srgrz 13861

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 2207	. . . . . . 7 |- (mulGrp ` R ) = (mulGrp ` R )
3		eqid 2207	. . . . . . 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 13842	. . . . . 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 1017	. . . . 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 2596	. . . 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 2581	. 2 |- ( R e. SRing -> A. x e. B ( x .x. .0. ) = .0. )
11		oveq1 5974	. . . 4 |- ( x = X -> ( x .x. .0. ) = ( X .x. .0. ) )
12	11	eqeq1d 2216	. . 3 |- ( x = X -> ( ( x .x. .0. ) = .0. <-> ( X .x. .0. ) = .0. ) )
13	12	rspcv 2880	. 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 1373   e. wcel 2178   A.wral 2486   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   0gc0g 13203   Mndcmnd 13363  CMndccmn 13735  mulGrpcmgp 13797  SRingcsrg 13840

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mulr 13038  df-0g 13205  df-srg 13841

This theorem is referenced by:  srgisid  13863  srglmhm  13870