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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 2622 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2622 | . . . . . . 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 18507 | . . . . . 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 1078 | . . . . 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 2932 | . . . 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 797 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → (𝑥 · 0 ) = 0 ) |
10 | 9 | ralrimiva 2966 | . 2 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 ) |
11 | oveq1 6657 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 · 0 ) = (𝑋 · 0 )) | |
12 | 11 | eqeq1d 2624 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 · 0 ) = 0 ↔ (𝑋 · 0 ) = 0 )) |
13 | 12 | rspcv 3305 | . 2 ⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 · 0 ) = 0 → (𝑋 · 0 ) = 0 )) |
14 | 10, 13 | mpan9 486 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 0gc0g 16100 Mndcmnd 17294 CMndccmn 18193 mulGrpcmgp 18489 SRingcsrg 18505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-srg 18506 |
This theorem is referenced by: srgisid 18528 srglmhm 18535 slmdvs0 29778 |
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