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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 | β’ πΉ = (Scalarβπ) |
slmdacl.k | β’ πΎ = (BaseβπΉ) |
slmdacl.p | β’ + = (+gβπΉ) |
Ref | Expression |
---|---|
slmdacl | β’ ((π β SLMod β§ π β πΎ β§ π β πΎ) β (π + π) β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdacl.f | . . . 4 β’ πΉ = (Scalarβπ) | |
2 | 1 | slmdsrg 32855 | . . 3 β’ (π β SLMod β πΉ β SRing) |
3 | srgmnd 20092 | . . 3 β’ (πΉ β SRing β πΉ β Mnd) | |
4 | 2, 3 | syl 17 | . 2 β’ (π β SLMod β πΉ β Mnd) |
5 | slmdacl.k | . . 3 β’ πΎ = (BaseβπΉ) | |
6 | slmdacl.p | . . 3 β’ + = (+gβπΉ) | |
7 | 5, 6 | mndcl 18672 | . 2 β’ ((πΉ β Mnd β§ π β πΎ β§ π β πΎ) β (π + π) β πΎ) |
8 | 4, 7 | syl3an1 1160 | 1 β’ ((π β SLMod β§ π β πΎ β§ π β πΎ) β (π + π) β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 Basecbs 17150 +gcplusg 17203 Scalarcsca 17206 Mndcmnd 18664 SRingcsrg 20088 SLModcslmd 32848 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6488 df-fv 6544 df-ov 7407 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-cmn 19699 df-srg 20089 df-slmd 32849 |
This theorem is referenced by: (None) |
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