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Mirrors > Home > MPE Home > Th. List > srgrz | Structured version Visualization version GIF version |
Description: The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgz.b | ⊢ 𝐵 = (Base‘𝑅) |
srgz.t | ⊢ · = (.r‘𝑅) |
srgz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
srgrz | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgz.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2724 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2724 | . . . . . . 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 20085 | . . . . . 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 1144 | . . . . 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 3240 | . . . 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 771 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → (𝑥 · 0 ) = 0 ) |
10 | 9 | ralrimiva 3138 | . 2 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 ) |
11 | oveq1 7409 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 · 0 ) = (𝑋 · 0 )) | |
12 | 11 | eqeq1d 2726 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 · 0 ) = 0 ↔ (𝑋 · 0 ) = 0 )) |
13 | 12 | rspcv 3600 | . 2 ⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 → (𝑋 · 0 ) = 0 )) |
14 | 10, 13 | mpan9 506 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ‘cfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 .rcmulr 17199 0gc0g 17386 Mndcmnd 18659 CMndccmn 19692 mulGrpcmgp 20031 SRingcsrg 20083 |
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 2163 ax-ext 2695 ax-nul 5297 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-ov 7405 df-srg 20084 |
This theorem is referenced by: srgisid 20106 srglmhm 20118 slmdvs0 32841 |
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