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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 · = (.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 3136 . . 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 20179 . . . 4 (𝑅 ∈ SRing → 0𝐵)
7 oveq2 7424 . . . . . 6 (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
87eqeq1d 2728 . . . . 5 (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
98rspcv 3603 . . . 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 20186 . . 3 ((𝑅 ∈ SRing ∧ 𝑍𝐵) → (𝑍 · 0 ) = 0 )
153, 12, 14syl2anc 582 . 2 (𝜑 → (𝑍 · 0 ) = 0 )
1611, 15eqtr3d 2768 1 (𝜑𝑍 = 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  0gc0g 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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