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Theorem srgisid 20227
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 · = (.r𝑅)
srgz.z 0 = (0g𝑅)
srgisid.1 (𝜑𝑅 ∈ SRing)
srgisid.2 (𝜑𝑍𝐵)
srgisid.3 ((𝜑𝑥𝐵) → (𝑍 · 𝑥) = 𝑍)
Assertion
Ref Expression
srgisid (𝜑𝑍 = 0 )
Distinct variable groups:   𝑥,𝐵   𝑥,𝑅   𝑥, ·   𝑥, 0   𝑥,𝑍   𝜑,𝑥

Proof of Theorem srgisid
StepHypRef Expression
1 srgisid.3 . . . 4 ((𝜑𝑥𝐵) → (𝑍 · 𝑥) = 𝑍)
21ralrimiva 3144 . . 3 (𝜑 → ∀𝑥𝐵 (𝑍 · 𝑥) = 𝑍)
3 srgisid.1 . . . 4 (𝜑𝑅 ∈ SRing)
4 srgz.b . . . . 5 𝐵 = (Base‘𝑅)
5 srgz.z . . . . 5 0 = (0g𝑅)
64, 5srg0cl 20218 . . . 4 (𝑅 ∈ SRing → 0𝐵)
7 oveq2 7439 . . . . . 6 (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
87eqeq1d 2737 . . . . 5 (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
98rspcv 3618 . . . 4 ( 0𝐵 → (∀𝑥𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
103, 6, 93syl 18 . . 3 (𝜑 → (∀𝑥𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
112, 10mpd 15 . 2 (𝜑 → (𝑍 · 0 ) = 𝑍)
12 srgisid.2 . . 3 (𝜑𝑍𝐵)
13 srgz.t . . . 4 · = (.r𝑅)
144, 13, 5srgrz 20225 . . 3 ((𝑅 ∈ SRing ∧ 𝑍𝐵) → (𝑍 · 0 ) = 0 )
153, 12, 14syl2anc 584 . 2 (𝜑 → (𝑍 · 0 ) = 0 )
1611, 15eqtr3d 2777 1 (𝜑𝑍 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  0gc0g 17486  SRingcsrg 20204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-cmn 19815  df-srg 20205
This theorem is referenced by: (None)
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