Theorem srgisid 14023

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 = (0g‘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 2604	. . 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 = (0g‘𝑅)
6	4, 5	srg0cl 14014	. . . 4 ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵)
7		oveq2 6031	. . . . . 6 ⊢ (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
8	7	eqeq1d 2239	. . . . 5 ⊢ (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
9	8	rspcv 2905	. . . 4 ⊢ ( 0 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
10	3, 6, 9	3syl 17	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
11	2, 10	mpd 13	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 𝑍)
12		srgisid.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13		srgz.t	. . . 4 ⊢ · = (.r‘𝑅)
14	4, 13, 5	srgrz 14021	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑍 ∈ 𝐵) → (𝑍 · 0 ) = 0 )
15	3, 12, 14	syl2anc 411	. 2 ⊢ (𝜑 → (𝑍 · 0 ) = 0 )
16	11, 15	eqtr3d 2265	1 ⊢ (𝜑 → 𝑍 = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  ._rcmulr 13184  0_gc0g 13362  SRingcsrg 14000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-riota 5976  df-ov 6026  df-inn 9149  df-2 9207  df-3 9208  df-ndx 13108  df-slot 13109  df-base 13111  df-plusg 13196  df-mulr 13197  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-cmn 13896  df-srg 14001

This theorem is referenced by: (None)