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Mirrors > Home > ILE Home > Th. List > srglz | GIF version |
Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.) |
Ref | Expression |
---|---|
srgz.b | ⊢ 𝐵 = (Base‘𝑅) |
srgz.t | ⊢ · = (.r‘𝑅) |
srgz.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
srglz | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgz.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2177 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2177 | . . . . . . 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 12961 | . . . . . 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 1014 | . . . . 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 2565 | . . . 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 530 | . . 3 ⊢ ((𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵) → ( 0 · 𝑥) = 0 ) |
10 | 9 | ralrimiva 2550 | . 2 ⊢ (𝑅 ∈ SRing → ∀𝑥 ∈ 𝐵 ( 0 · 𝑥) = 0 ) |
11 | oveq2 5876 | . . . 4 ⊢ (𝑥 = 𝑋 → ( 0 · 𝑥) = ( 0 · 𝑋)) | |
12 | 11 | eqeq1d 2186 | . . 3 ⊢ (𝑥 = 𝑋 → (( 0 · 𝑥) = 0 ↔ ( 0 · 𝑋) = 0 )) |
13 | 12 | rspcv 2837 | . 2 ⊢ (𝑋 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ( 0 · 𝑥) = 0 → ( 0 · 𝑋) = 0 )) |
14 | 10, 13 | mpan9 281 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 .rcmulr 12506 0gc0g 12640 Mndcmnd 12696 CMndccmn 12902 mulGrpcmgp 12944 SRingcsrg 12959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-3 8955 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-mulr 12519 df-0g 12642 df-srg 12960 |
This theorem is referenced by: srgmulgass 12985 srgrmhm 12990 |
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