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Mirrors > Home > MPE Home > Th. List > lmod0vid | Structured version Visualization version GIF version |
Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
0vlid.v | ⊢ 𝑉 = (Base‘𝑊) |
0vlid.a | ⊢ + = (+g‘𝑊) |
0vlid.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lmod0vid | ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodgrp 20282 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
2 | 0vlid.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 0vlid.a | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 0vlid.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | 2, 3, 4 | grpid 18746 | . 2 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
6 | 1, 5 | sylan 581 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 +gcplusg 17093 0gc0g 17281 Grpcgrp 18708 LModclmod 20275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-riota 7308 df-ov 7355 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-lmod 20277 |
This theorem is referenced by: lmod0vs 20308 dva0g 39422 dvh0g 39506 |
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