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Theorem srgisid 18292
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 2943 . . 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 18283 . . . 4 (𝑅 ∈ SRing → 0𝐵)
7 oveq2 6530 . . . . . 6 (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
87eqeq1d 2606 . . . . 5 (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
98rspcv 3272 . . . 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 18290 . . 3 ((𝑅 ∈ SRing ∧ 𝑍𝐵) → (𝑍 · 0 ) = 0 )
153, 12, 14syl2anc 690 . 2 (𝜑 → (𝑍 · 0 ) = 0 )
1611, 15eqtr3d 2640 1 (𝜑𝑍 = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  wral 2890  cfv 5785  (class class class)co 6522  Basecbs 15636  .rcmulr 15710  0gc0g 15864  SRingcsrg 18269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-iota 5749  df-fun 5787  df-fv 5793  df-riota 6484  df-ov 6525  df-0g 15866  df-mgm 17006  df-sgrp 17048  df-mnd 17059  df-cmn 17959  df-srg 18270
This theorem is referenced by: (None)
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