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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 32935 | . . 3 β’ (π β SLMod β πΉ β SRing) |
3 | srgmnd 20137 | . . 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 18709 | . 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 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 Scalarcsca 17243 Mndcmnd 18701 SRingcsrg 20133 SLModcslmd 32928 |
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 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-cmn 19744 df-srg 20134 df-slmd 32929 |
This theorem is referenced by: (None) |
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