Theorem srgisid 14290

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 2623	. . 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 14281	. . . 4 |- ( R e. SRing -> .0. e. B )
7		oveq2 6093	. . . . . 6 |- ( x = .0. -> ( Z .x. x ) = ( Z .x. .0. ) )
8	7	eqeq1d 2247	. . . . 5 |- ( x = .0. -> ( ( Z .x. x ) = Z <-> ( Z .x. .0. ) = Z ) )
9	8	rspcv 2925	. . . 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 14288	. . 3 |- ( ( R e. SRing /\ Z e. B ) -> ( Z .x. .0. ) = .0. )
15	3, 12, 14	syl2anc 415	. 2 |- ( ph -> ( Z .x. .0. ) = .0. )
16	11, 15	eqtr3d 2273	1 |- ( ph -> Z = .0. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   ` cfv 5377  (class class class)co 6085   Basecbs 13352   .rcmulr 13432   0gc0g 13610  SRingcsrg 14267

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-cmn 14089  df-srg 14268

This theorem is used by: (None)