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Theorem srglz 19273
 Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgz.b 𝐵 = (Base‘𝑅)
srgz.t · = (.r𝑅)
srgz.z 0 = (0g𝑅)
Assertion
Ref Expression
srglz ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )

Proof of Theorem srglz
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 2798 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2798 . . . . . . 7 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . 7 · = (.r𝑅)
5 srgz.z . . . . . . 7 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 19253 . . . . . 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 3173 . . . 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 )))
98simprld 771 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → ( 0 · 𝑥) = 0 )
109ralrimiva 3149 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 ( 0 · 𝑥) = 0 )
11 oveq2 7143 . . . 4 (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋))
1211eqeq1d 2800 . . 3 (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 ))
1312rspcv 3566 . 2 (𝑋𝐵 → (∀𝑥𝐵 ( 0 · 𝑥) = 0 → ( 0 · 𝑋) = 0 ))
1410, 13mpan9 510 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  .rcmulr 16560  0gc0g 16707  Mndcmnd 17905  CMndccmn 18901  mulGrpcmgp 19235  SRingcsrg 19251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-srg 19252 This theorem is referenced by:  srgmulgass  19277  srgrmhm  19282
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