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Theorem srglz 19542
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 2737 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2737 . . . . . . 7 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . 7 · = (.r𝑅)
5 srgz.z . . . . . . 7 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 19522 . . . . . 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 1149 . . . . 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 3130 . . . 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 772 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → ( 0 · 𝑥) = 0 )
109ralrimiva 3105 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 ( 0 · 𝑥) = 0 )
11 oveq2 7221 . . . 4 (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋))
1211eqeq1d 2739 . . 3 (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 ))
1312rspcv 3532 . 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 1543  wcel 2110  wral 3061  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  .rcmulr 16803  0gc0g 16944  Mndcmnd 18173  CMndccmn 19170  mulGrpcmgp 19504  SRingcsrg 19520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-srg 19521
This theorem is referenced by:  srgmulgass  19546  srgrmhm  19551
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