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Theorem srglz 20155
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 2728 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2728 . . . . . . 7 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . 7 · = (.r𝑅)
5 srgz.z . . . . . . 7 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 20135 . . . . . 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 3246 . . . 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 770 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → ( 0 · 𝑥) = 0 )
109ralrimiva 3143 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 ( 0 · 𝑥) = 0 )
11 oveq2 7434 . . . 4 (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋))
1211eqeq1d 2730 . . 3 (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 ))
1312rspcv 3607 . 2 (𝑋𝐵 → (∀𝑥𝐵 ( 0 · 𝑥) = 0 → ( 0 · 𝑋) = 0 ))
1410, 13mpan9 505 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  .rcmulr 17241  0gc0g 17428  Mndcmnd 18701  CMndccmn 19742  mulGrpcmgp 20081  SRingcsrg 20133
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 2166  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-srg 20134
This theorem is referenced by:  srgmulgass  20164  srgrmhm  20169
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