Theorem srglz 19763

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	⊢ · = (.r‘𝑅)
srgz.z	⊢ 0 = (0g‘𝑅)

Assertion

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

Proof of Theorem srglz

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 2738	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
3		eqid 2738	. . . . . . 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 19743	. . . . . 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 1146	. . . . 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 3134	. . . 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	simprld 769	. . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 )
10	9	ralrimiva 3103	. 2 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 ( 0 · 𝑥) = 0 )
11		oveq2 7283	. . . 4 ⊢ (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋))
12	11	eqeq1d 2740	. . 3 ⊢ (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 ))
13	12	rspcv 3557	. 2 ⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ( 0 · 𝑥) = 0 → ( 0 · 𝑋) = 0 ))
14	10, 13	mpan9 507	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  ._rcmulr 16963  0_gc0g 17150  Mndcmnd 18385  CMndccmn 19386  mulGrpcmgp 19720  SRingcsrg 19741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-srg 19742

This theorem is referenced by:  srgmulgass  19767  srgrmhm  19772