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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 18920 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
3 | lmodacl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | lmodacl.p | . . 3 ⊢ + = (+g‘𝐹) | |
5 | 3, 4 | grpcl 17477 | . 2 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
6 | 2, 5 | syl3an1 1399 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 Scalarcsca 15991 Grpcgrp 17469 LModclmod 18911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-nul 4822 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-iota 5889 df-fv 5934 df-ov 6693 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-ring 18595 df-lmod 18913 |
This theorem is referenced by: lmodcom 18957 lss1d 19011 lspsolvlem 19190 lfladdcl 34676 lshpkrlem5 34719 ldualvsdi2 34749 baerlem5blem1 37315 hgmapadd 37503 |
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