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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmd0cl | Structured version Visualization version GIF version |
Description: The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmd0cl.f | β’ πΉ = (Scalarβπ) |
slmd0cl.k | β’ πΎ = (BaseβπΉ) |
slmd0cl.z | β’ 0 = (0gβπΉ) |
Ref | Expression |
---|---|
slmd0cl | β’ (π β SLMod β 0 β πΎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmd0cl.f | . . 3 β’ πΉ = (Scalarβπ) | |
2 | 1 | slmdsrg 32855 | . 2 β’ (π β SLMod β πΉ β SRing) |
3 | slmd0cl.k | . . 3 β’ πΎ = (BaseβπΉ) | |
4 | slmd0cl.z | . . 3 β’ 0 = (0gβπΉ) | |
5 | 3, 4 | srg0cl 20102 | . 2 β’ (πΉ β SRing β 0 β πΎ) |
6 | 2, 5 | syl 17 | 1 β’ (π β SLMod β 0 β πΎ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βcfv 6536 Basecbs 17150 Scalarcsca 17206 0gc0g 17391 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-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 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-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-ov 7407 df-0g 17393 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-cmn 19699 df-srg 20089 df-slmd 32849 |
This theorem is referenced by: slmd0vs 32872 |
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