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Theorem srgrz 20104
Description: The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgz.b 𝐵 = (Base‘𝑅)
srgz.t · = (.r𝑅)
srgz.z 0 = (0g𝑅)
Assertion
Ref Expression
srgrz ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )

Proof of Theorem srgrz
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgz.b . . . . . . 7 𝐵 = (Base‘𝑅)
2 eqid 2724 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2724 . . . . . . 7 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . 7 · = (.r𝑅)
5 srgz.z . . . . . . 7 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 20085 . . . . . 6 (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
76simp3bi 1144 . . . . 5 (𝑅 ∈ SRing → ∀𝑥𝐵 (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
87r19.21bi 3240 . . . 4 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (∀𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦(+g𝑅)𝑧)) = ((𝑥 · 𝑦)(+g𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
98simprrd 771 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (𝑥 · 0 ) = 0 )
109ralrimiva 3138 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 (𝑥 · 0 ) = 0 )
11 oveq1 7409 . . . 4 (𝑥 = 𝑋 → (𝑥 · 0 ) = (𝑋 · 0 ))
1211eqeq1d 2726 . . 3 (𝑥 = 𝑋 → ((𝑥 · 0 ) = 0 ↔ (𝑋 · 0 ) = 0 ))
1312rspcv 3600 . 2 (𝑋𝐵 → (∀𝑥𝐵 (𝑥 · 0 ) = 0 → (𝑋 · 0 ) = 0 ))
1410, 13mpan9 506 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  cfv 6534  (class class class)co 7402  Basecbs 17145  +gcplusg 17198  .rcmulr 17199  0gc0g 17386  Mndcmnd 18659  CMndccmn 19692  mulGrpcmgp 20031  SRingcsrg 20083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-srg 20084
This theorem is referenced by:  srgisid  20106  srglmhm  20118  slmdvs0  32841
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