srgisid - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > srgisid		Structured version Visualization version GIF version

Theorem srgisid 20125

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	⊢ · = (._r‘𝑅)
srgz.z	⊢ 0 = (0_g‘𝑅)
srgisid.1	⊢ (𝜑 → 𝑅 ∈ SRing)
srgisid.2	⊢ (𝜑 → 𝑍 ∈ 𝐵)
srgisid.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍)

**Assertion**
Ref	Expression
srgisid	⊢ (𝜑 → 𝑍 = 0 )

Distinct variable groups: 𝑥,𝐵 𝑥,𝑅 𝑥, · 𝑥, 0 𝑥,𝑍 𝜑,𝑥

**Proof of Theorem srgisid**
Step	Hyp	Ref	Expression
1		srgisid.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍)
2	1	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍)
3		srgisid.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ SRing)
4		srgz.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
5		srgz.z	. . . . 5 ⊢ 0 = (0_g‘𝑅)
6	4, 5	srg0cl 20116	. . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵)
7		oveq2 7354	. . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
8	7	eqeq1d 2733	. . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
9	8	rspcv 3573	. . . 4 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
10	3, 6, 9	3syl 18	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
11	2, 10	mpd 15	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 𝑍)
12		srgisid.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13		srgz.t	. . . 4 ⊢ · = (._r‘𝑅)
14	4, 13, 5	srgrz 20123	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 )
15	3, 12, 14	syl2anc 584	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 )
16	11, 15	eqtr3d 2768	1 ⊢ (𝜑 → 𝑍 = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ‘cfv 6481 (class class class)co 7346 Basecbs 17117 ._rcmulr 17159 0_gc0g 17340 SRingcsrg 20102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-iota 6437 df-fun 6483 df-fv 6489 df-riota 7303 df-ov 7349 df-0g 17342 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-cmn 19692 df-srg 20103

This theorem is referenced by: (None)

W3C validator