Theorem srgisid 13485

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 2567	. . 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 13476	. . . 4 |- ( R e. SRing -> .0. e. B )
7		oveq2 5927	. . . . . 6 |- ( x = .0. -> ( Z .x. x ) = ( Z .x. .0. ) )
8	7	eqeq1d 2202	. . . . 5 |- ( x = .0. -> ( ( Z .x. x ) = Z <-> ( Z .x. .0. ) = Z ) )
9	8	rspcv 2861	. . . 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 13483	. . 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 2228	1 |- ( ph -> Z = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   ` cfv 5255  (class class class)co 5919   Basecbs 12621   .rcmulr 12699   0gc0g 12870  SRingcsrg 13462

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-cmn 13359  df-srg 13463

This theorem is referenced by: (None)