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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdacl | Structured version Visualization version GIF version |
Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdacl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
slmdacl.k | ⊢ 𝐾 = (Base‘𝐹) |
slmdacl.p | ⊢ + = (+g‘𝐹) |
Ref | Expression |
---|---|
slmdacl | ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdacl.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | slmdsrg 30885 | . . 3 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | srgmnd 19252 | . . 3 ⊢ (𝐹 ∈ SRing → 𝐹 ∈ Mnd) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ Mnd) |
5 | slmdacl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
6 | slmdacl.p | . . 3 ⊢ + = (+g‘𝐹) | |
7 | 5, 6 | mndcl 17911 | . 2 ⊢ ((𝐹 ∈ Mnd ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
8 | 4, 7 | syl3an1 1160 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 Mndcmnd 17903 SRingcsrg 19248 SLModcslmd 30878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cmn 18900 df-srg 19249 df-slmd 30879 |
This theorem is referenced by: (None) |
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