Theorem srgisid 12962

Description: In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.)

Hypotheses

Ref	Expression
srgz.b	|- B = ( Base ` R )
srgz.t	|- .x. = ( .r ` R )
srgz.z	|- .0. = ( 0g ` R )
srgisid.1	|- ( ph -> R e. SRing )
srgisid.2	|- ( ph -> Z e. B )
srgisid.3	|- ( ( ph /\ x e. B ) -> ( Z .x. x ) = Z )

Assertion

Ref	Expression
srgisid	|- ( ph -> Z = .0. )

Distinct variable groups:   x, B   x, R   x, .x.   x, .0.   x, Z   ph, x

Proof of Theorem srgisid

Step	Hyp	Ref	Expression
1		srgisid.3	. . . 4 |- ( ( ph /\ x e. B ) -> ( Z .x. x ) = Z )
2	1	ralrimiva 2548	. . 3 |- ( ph -> A. x e. B ( Z .x. x ) = Z )
3		srgisid.1	. . . 4 |- ( ph -> R e. SRing )
4		srgz.b	. . . . 5 |- B = ( Base ` R )
5		srgz.z	. . . . 5 |- .0. = ( 0g ` R )
6	4, 5	srg0cl 12953	. . . 4 |- ( R e. SRing -> .0. e. B )
7		oveq2 5873	. . . . . 6 |- ( x = .0. -> ( Z .x. x ) = ( Z .x. .0. ) )
8	7	eqeq1d 2184	. . . . 5 |- ( x = .0. -> ( ( Z .x. x ) = Z <-> ( Z .x. .0. ) = Z ) )
9	8	rspcv 2835	. . . 4 |- ( .0. e. B -> ( A. x e. B ( Z .x. x ) = Z -> ( Z .x. .0. ) = Z ) )
10	3, 6, 9	3syl 17	. . 3 |- ( ph -> ( A. x e. B ( Z .x. x ) = Z -> ( Z .x. .0. ) = Z ) )
11	2, 10	mpd 13	. 2 |- ( ph -> ( Z .x. .0. ) = Z )
12		srgisid.2	. . 3 |- ( ph -> Z e. B )
13		srgz.t	. . . 4 |- .x. = ( .r ` R )
14	4, 13, 5	srgrz 12960	. . 3 |- ( ( R e. SRing /\ Z e. B ) -> ( Z .x. .0. ) = .0. )
15	3, 12, 14	syl2anc 411	. 2 |- ( ph -> ( Z .x. .0. ) = .0. )
16	11, 15	eqtr3d 2210	1 |- ( ph -> Z = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   A.wral 2453   ` cfv 5208  (class class class)co 5865   Basecbs 12428   .rcmulr 12493   0gc0g 12626  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-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  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-mgm 12640  df-sgrp 12673  df-mnd 12683  df-cmn 12886  df-srg 12940

This theorem is referenced by: (None)