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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmd0vcl | Structured version Visualization version GIF version |
Description: The zero vector is a vector. (ax-hv0cl 28786 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmd0vcl.v | ⊢ 𝑉 = (Base‘𝑊) |
slmd0vcl.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
slmd0vcl | ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdmnd 30884 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmd0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | slmd0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | mndidcl 17918 | . 2 ⊢ (𝑊 ∈ Mnd → 0 ∈ 𝑉) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 Basecbs 16475 0gc0g 16705 Mndcmnd 17903 SLModcslmd 30878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-riota 7093 df-ov 7138 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-cmn 18900 df-slmd 30879 |
This theorem is referenced by: slmdvs0 30903 |
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