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Theorem srgrz 19276
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 2824 . . . . . . 7 (mulGrp‘𝑅) = (mulGrp‘𝑅)
3 eqid 2824 . . . . . . 7 (+g𝑅) = (+g𝑅)
4 srgz.t . . . . . . 7 · = (.r𝑅)
5 srgz.z . . . . . . 7 0 = (0g𝑅)
61, 2, 3, 4, 5issrg 19257 . . . . . 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 3203 . . . 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 773 . . 3 ((𝑅 ∈ SRing ∧ 𝑥𝐵) → (𝑥 · 0 ) = 0 )
109ralrimiva 3177 . 2 (𝑅 ∈ SRing → ∀𝑥𝐵 (𝑥 · 0 ) = 0 )
11 oveq1 7156 . . . 4 (𝑥 = 𝑋 → (𝑥 · 0 ) = (𝑋 · 0 ))
1211eqeq1d 2826 . . 3 (𝑥 = 𝑋 → ((𝑥 · 0 ) = 0 ↔ (𝑋 · 0 ) = 0 ))
1312rspcv 3604 . 2 (𝑋𝐵 → (∀𝑥𝐵 (𝑥 · 0 ) = 0 → (𝑋 · 0 ) = 0 ))
1410, 13mpan9 510 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  0gc0g 16713  Mndcmnd 17911  CMndccmn 18906  mulGrpcmgp 19239  SRingcsrg 19255
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-srg 19256
This theorem is referenced by:  srgisid  19278  srglmhm  19285  slmdvs0  30885
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