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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 20884 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
3 | lmodacl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmodacl.p | . . 3 ⊢ + = (+g‘𝐹) | |
5 | 3, 4 | grpcl 18972 | . 2 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
6 | 2, 5 | syl3an1 1162 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 Grpcgrp 18964 LModclmod 20875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-ring 20253 df-lmod 20877 |
This theorem is referenced by: lmodcom 20923 lss1d 20979 lspsolvlem 21162 lmodvslmhm 33036 imaslmod 33361 lfladdcl 39053 lshpkrlem5 39096 ldualvsdi2 39126 baerlem5blem1 41692 hgmapadd 41877 |
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