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Mirrors > Home > MPE Home > Th. List > lmodacl | Structured version Visualization version GIF version |
Description: Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodacl.f | ⊢ 𝐹 = (Scalar‘𝑊) |
lmodacl.k | ⊢ 𝐾 = (Base‘𝐹) |
lmodacl.p | ⊢ + = (+g‘𝐹) |
Ref | Expression |
---|---|
lmodacl | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodacl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | lmodfgrp 19941 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
3 | lmodacl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmodacl.p | . . 3 ⊢ + = (+g‘𝐹) | |
5 | 3, 4 | grpcl 18406 | . 2 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
6 | 2, 5 | syl3an1 1165 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ‘cfv 6401 (class class class)co 7235 Basecbs 16793 +gcplusg 16835 Scalarcsca 16838 Grpcgrp 18398 LModclmod 19932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-nul 5216 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-iota 6359 df-fv 6409 df-ov 7238 df-mgm 18147 df-sgrp 18196 df-mnd 18207 df-grp 18401 df-ring 19597 df-lmod 19934 |
This theorem is referenced by: lmodcom 19978 lss1d 20033 lspsolvlem 20212 lmodvslmhm 31061 imaslmod 31299 lfladdcl 36859 lshpkrlem5 36902 ldualvsdi2 36932 baerlem5blem1 39497 hgmapadd 39682 |
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