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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 32908 | . . 3 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | srgmnd 20123 | . . 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 18695 | . 2 ⊢ ((𝐹 ∈ Mnd ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
8 | 4, 7 | syl3an1 1161 | 1 ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 Scalarcsca 17229 Mndcmnd 18687 SRingcsrg 20119 SLModcslmd 32901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-nul 5300 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-cmn 19730 df-srg 20120 df-slmd 32902 |
This theorem is referenced by: (None) |
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