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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 30765 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 32857 | . 2 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | |
2 | slmd0vcl.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | slmd0vcl.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
4 | 2, 3 | mndidcl 18682 | . 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 1533 ∈ wcel 2098 ‘cfv 6537 Basecbs 17153 0gc0g 17394 Mndcmnd 18667 SLModcslmd 32851 |
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 6489 df-fun 6539 df-fv 6545 df-riota 7361 df-ov 7408 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-cmn 19702 df-slmd 32852 |
This theorem is referenced by: slmdvs0 32876 |
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