Theorem srgisid 13174

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 2550	. . 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 13165	. . . 4 |- ( R e. SRing -> .0. e. B )
7		oveq2 5885	. . . . . 6 |- ( x = .0. -> ( Z .x. x ) = ( Z .x. .0. ) )
8	7	eqeq1d 2186	. . . . 5 |- ( x = .0. -> ( ( Z .x. x ) = Z <-> ( Z .x. .0. ) = Z ) )
9	8	rspcv 2839	. . . 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 13172	. . 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 2212	1 |- ( ph -> Z = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   ` cfv 5218  (class class class)co 5877   Basecbs 12464   .rcmulr 12539   0gc0g 12710  SRingcsrg 13151

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mulr 12552  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-cmn 13095  df-srg 13152

This theorem is referenced by: (None)