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Theorem slmd0cl 28535
Description: The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0cl.f  |-  F  =  (Scalar `  W )
slmd0cl.k  |-  K  =  ( Base `  F
)
slmd0cl.z  |-  .0.  =  ( 0g `  F )
Assertion
Ref Expression
slmd0cl  |-  ( W  e. SLMod  ->  .0.  e.  K
)

Proof of Theorem slmd0cl
StepHypRef Expression
1 slmd0cl.f . . 3  |-  F  =  (Scalar `  W )
21slmdsrg 28524 . 2  |-  ( W  e. SLMod  ->  F  e. SRing )
3 slmd0cl.k . . 3  |-  K  =  ( Base `  F
)
4 slmd0cl.z . . 3  |-  .0.  =  ( 0g `  F )
53, 4srg0cl 17745 . 2  |-  ( F  e. SRing  ->  .0.  e.  K
)
62, 5syl 17 1  |-  ( W  e. SLMod  ->  .0.  e.  K
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869   ` cfv 5599   Basecbs 15114  Scalarcsca 15186   0gc0g 15331  SRingcsrg 17732  SLModcslmd 28517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-riota 6265  df-ov 6306  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-cmn 17425  df-srg 17733  df-slmd 28518
This theorem is referenced by:  slmd0vs  28541
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