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Theorem lmod0vid 18661
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.)
Hypotheses
Ref Expression
0vlid.v 𝑉 = (Base‘𝑊)
0vlid.a + = (+g𝑊)
0vlid.z 0 = (0g𝑊)
Assertion
Ref Expression
lmod0vid ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))

Proof of Theorem lmod0vid
StepHypRef Expression
1 lmodgrp 18636 . 2 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
2 0vlid.v . . 3 𝑉 = (Base‘𝑊)
3 0vlid.a . . 3 + = (+g𝑊)
4 0vlid.z . . 3 0 = (0g𝑊)
52, 3, 4grpid 17223 . 2 ((𝑊 ∈ Grp ∧ 𝑋𝑉) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
61, 5sylan 486 1 ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  cfv 5787  (class class class)co 6524  Basecbs 15638  +gcplusg 15711  0gc0g 15866  Grpcgrp 17188  LModclmod 18629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795  df-riota 6486  df-ov 6527  df-0g 15868  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-grp 17191  df-lmod 18631
This theorem is referenced by:  lmod0vs  18662  dva0g  35134  dvh0g  35218
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