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Theorem slmd0vlid 32923
Description: Left identity law for the zero vector. (hvaddlid 30826 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vlid.v 𝑉 = (Base‘𝑊)
slmd0vlid.a + = (+g𝑊)
slmd0vlid.z 0 = (0g𝑊)
Assertion
Ref Expression
slmd0vlid ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)

Proof of Theorem slmd0vlid
StepHypRef Expression
1 slmdmnd 32907 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vlid.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vlid.a . . 3 + = (+g𝑊)
4 slmd0vlid.z . . 3 0 = (0g𝑊)
52, 3, 4mndlid 18707 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)
61, 5sylan 579 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  Basecbs 17173  +gcplusg 17226  0gc0g 17414  Mndcmnd 18687  SLModcslmd 32901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7370  df-ov 7417  df-0g 17416  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-cmn 19730  df-slmd 32902
This theorem is referenced by: (None)
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