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Theorem srgisid 12982
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 2550 . . 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 12973 . . . 4 (𝑅 ∈ SRing → 0𝐵)
7 oveq2 5876 . . . . . 6 (𝑥 = 0 → (𝑍 · 𝑥) = (𝑍 · 0 ))
87eqeq1d 2186 . . . . 5 (𝑥 = 0 → ((𝑍 · 𝑥) = 𝑍 ↔ (𝑍 · 0 ) = 𝑍))
98rspcv 2837 . . . 4 ( 0𝐵 → (∀𝑥𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
103, 6, 93syl 17 . . 3 (𝜑 → (∀𝑥𝐵 (𝑍 · 𝑥) = 𝑍 → (𝑍 · 0 ) = 𝑍))
112, 10mpd 13 . 2 (𝜑 → (𝑍 · 0 ) = 𝑍)
12 srgisid.2 . . 3 (𝜑𝑍𝐵)
13 srgz.t . . . 4 · = (.r𝑅)
144, 13, 5srgrz 12980 . . 3 ((𝑅 ∈ SRing ∧ 𝑍𝐵) → (𝑍 · 0 ) = 0 )
153, 12, 14syl2anc 411 . 2 (𝜑 → (𝑍 · 0 ) = 0 )
1611, 15eqtr3d 2212 1 (𝜑𝑍 = 0 )
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  cfv 5211  (class class class)co 5868  Basecbs 12432  .rcmulr 12506  0gc0g 12640  SRingcsrg 12959
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-riota 5824  df-ov 5871  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-cmn 12904  df-srg 12960
This theorem is referenced by: (None)
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