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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 28432 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 30321 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmd0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | slmd0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | mndidcl 17694 | . 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 1601 ∈ wcel 2107 ‘cfv 6135 Basecbs 16255 0gc0g 16486 Mndcmnd 17680 SLModcslmd 30315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-riota 6883 df-ov 6925 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-cmn 18581 df-slmd 30316 |
This theorem is referenced by: slmdvs0 30340 |
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