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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 29091 | . . 3 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | srgmnd 18332 | . . 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 17124 | . 2 ⊢ ((𝐹 ∈ Mnd ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
8 | 4, 7 | syl3an1 1351 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 Mndcmnd 17117 SRingcsrg 18328 SLModcslmd 29084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-cmn 18018 df-srg 18329 df-slmd 29085 |
This theorem is referenced by: (None) |
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