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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 | ⊢ 𝐹 = (Scalar‘𝑊) |
slmd0cl.k | ⊢ 𝐾 = (Base‘𝐹) |
slmd0cl.z | ⊢ 0 = (0g‘𝐹) |
Ref | Expression |
---|---|
slmd0cl | ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmd0cl.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | slmdsrg 31179 | . 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
3 | slmd0cl.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
4 | slmd0cl.z | . . 3 ⊢ 0 = (0g‘𝐹) | |
5 | 3, 4 | srg0cl 19534 | . 2 ⊢ (𝐹 ∈ SRing → 0 ∈ 𝐾) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 Basecbs 16760 Scalarcsca 16805 0gc0g 16944 SRingcsrg 19520 SLModcslmd 31172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-riota 7170 df-ov 7216 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-cmn 19172 df-srg 19521 df-slmd 31173 |
This theorem is referenced by: slmd0vs 31196 |
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