Theorem srg0cl 18519

Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srg0cl.b	|- B = ( Base ` R )
srg0cl.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
srg0cl	|- ( R e. SRing -> .0. e. B )

Proof of Theorem srg0cl

Step	Hyp	Ref	Expression
1		srgmnd 18509	. 2 |- ( R e. SRing -> R e. Mnd )
2		srg0cl.b	. . 3 |- B = ( Base ` R )
3		srg0cl.z	. . 3 |- .0. = ( 0g ` R )
4	2, 3	mndidcl 17308	. 2 |- ( R e. Mnd -> .0. e. B )
5	1, 4	syl 17	1 |- ( R e. SRing -> .0. e. B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   Basecbs 15857   0gc0g 16100   Mndcmnd 17294  SRingcsrg 18505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195  df-srg 18506

This theorem is referenced by:  srgisid  18528  srgen1zr  18530  srglmhm  18535  srgrmhm  18536  slmd0cl  29771  slmdvs0  29778