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Theorem srgisid 20354

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 3156	. . 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 20345	. . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵)
7		oveq2 7425	. . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
8	7	eqeq1d 2764	. . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
9	8	rspcv 3575	. . . 4 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
10	3, 6, 9	3syl 19	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
11	2, 10	mpd 16	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 𝑍)
12		srgisid.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13		srgz.t	. . . 4 ⊢ · = (._r‘𝑅)
14	4, 13, 5	srgrz 20352	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 )
15	3, 12, 14	syl2anc 596	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 )
16	11, 15	eqtr3d 2799	1 ⊢ (𝜑 → 𝑍 = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3078 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 ._rcmulr 17349 0_gc0g 17530 SRingcsrg 20331

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-iota 6493 df-fun 6539 df-fv 6545 df-riota 7374 df-ov 7420 df-0g 17532 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-cmn 19915 df-srg 20332

This theorem is used by: (None)

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