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Theorem srgisid 20188

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 3136	. . 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 20179	. . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵)
7		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
8	7	eqeq1d 2728	. . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
9	8	rspcv 3603	. . . 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 20186	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 )
15	3, 12, 14	syl2anc 582	. 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 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 ._rcmulr 17262 0_gc0g 17449 SRingcsrg 20165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-riota 7372 df-ov 7419 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-cmn 19776 df-srg 20166

This theorem is referenced by: (None)

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