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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 20867 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
| 3 | lmodacl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
| 4 | lmodacl.p | . . 3 ⊢ + = (+g‘𝐹) | |
| 5 | 3, 4 | grpcl 18959 | . 2 ⊢ ((𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
| 6 | 2, 5 | syl3an1 1164 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 Grpcgrp 18951 LModclmod 20858 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 df-lmod 20860 |
| This theorem is referenced by: lmodcom 20906 lss1d 20961 lspsolvlem 21144 lmodvslmhm 33053 imaslmod 33381 lfladdcl 39072 lshpkrlem5 39115 ldualvsdi2 39145 baerlem5blem1 41711 hgmapadd 41896 |
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