Theorem srgrz 20442

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	⊢ · = (.r‘𝑅)
srgz.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
srgrz	⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )

Proof of Theorem srgrz

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		srgz.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2760	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2760	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		srgz.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
5		srgz.z	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	issrg 20423	. . . . . 6 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 ))))
7	6	simp3bi 1165	. . . . 5 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
8	7	r19.21bi 3254	. . . 4 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦(+g‘𝑅)𝑧)) = ((𝑥 · 𝑦)(+g‘𝑅)(𝑥 · 𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦) · 𝑧) = ((𝑥 · 𝑧)(+g‘𝑅)(𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))
9	8	simprrd 786	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → (𝑥 · 0 ) = 0 )
10	9	ralrimiva 3154	. 2 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 )
11		oveq1 7431	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 · 0 ) = (𝑋 · 0 ))
12	11	eqeq1d 2762	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 · 0 ) = 0 ↔ (𝑋 · 0 ) = 0 ))
13	12	rspcv 3572	. 2 ⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 → (𝑋 · 0 ) = 0 ))
14	10, 13	mpan9 516	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  ._rcmulr 17438  0_gc0g 17619  Mndcmnd 18932  CMndccmn 20003  mulGrpcmgp 20369  SRingcsrg 20421

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-srg 20422

This theorem is used by:  srgisid  20444  srglmhm  20456  slmdvs0  33836  rhmqusspan  43344