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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 32935 | . 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 20147 | . 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 6553 Basecbs 17187 Scalarcsca 17243 0gc0g 17428 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-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 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-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-ov 7429 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-cmn 19744 df-srg 20134 df-slmd 32929 |
This theorem is referenced by: slmd0vs 32952 |
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