Theorem srgisid 13998

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 2605	. . 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 13989	. . . 4 |- ( R e. SRing -> .0. e. B )
7		oveq2 6025	. . . . . 6 |- ( x = .0. -> ( Z .x. x ) = ( Z .x. .0. ) )
8	7	eqeq1d 2240	. . . . 5 |- ( x = .0. -> ( ( Z .x. x ) = Z <-> ( Z .x. .0. ) = Z ) )
9	8	rspcv 2906	. . . 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 13996	. . 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 2266	1 |- ( ph -> Z = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   0gc0g 13338  SRingcsrg 13975

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-cmn 13872  df-srg 13976

This theorem is referenced by: (None)