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Theorem srgisid 20207
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 3145 . . 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 20198 . . . 4 (𝑅 ∈ SRing → 0𝐵)
7 oveq2 7440 . . . . . 6 (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
87eqeq1d 2738 . . . . 5 (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
98rspcv 3617 . . . 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 20205 . . 3 ((𝑅 ∈ SRing ∧ 𝑍𝐵) → (𝑍 · 0 ) = 0 )
153, 12, 14syl2anc 584 . 2 (𝜑 → (𝑍 · 0 ) = 0 )
1611, 15eqtr3d 2778 1 (𝜑𝑍 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  .rcmulr 17299  0gc0g 17485  SRingcsrg 20184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-cmn 19801  df-srg 20185
This theorem is referenced by: (None)
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