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Theorem lmod0vs 20370
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 29994 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmod0vs.v 𝑉 = (Baseβ€˜π‘Š)
lmod0vs.f 𝐹 = (Scalarβ€˜π‘Š)
lmod0vs.s Β· = ( ·𝑠 β€˜π‘Š)
lmod0vs.o 𝑂 = (0gβ€˜πΉ)
lmod0vs.z 0 = (0gβ€˜π‘Š)
Assertion
Ref Expression
lmod0vs ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) = 0 )

Proof of Theorem lmod0vs
StepHypRef Expression
1 simpl 484 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ π‘Š ∈ LMod)
2 lmod0vs.f . . . . . . . 8 𝐹 = (Scalarβ€˜π‘Š)
32lmodring 20344 . . . . . . 7 (π‘Š ∈ LMod β†’ 𝐹 ∈ Ring)
43adantr 482 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝐹 ∈ Ring)
5 eqid 2733 . . . . . . 7 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
6 lmod0vs.o . . . . . . 7 𝑂 = (0gβ€˜πΉ)
75, 6ring0cl 19995 . . . . . 6 (𝐹 ∈ Ring β†’ 𝑂 ∈ (Baseβ€˜πΉ))
84, 7syl 17 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑂 ∈ (Baseβ€˜πΉ))
9 simpr 486 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ 𝑉)
10 lmod0vs.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
11 eqid 2733 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
12 lmod0vs.s . . . . . 6 Β· = ( ·𝑠 β€˜π‘Š)
13 eqid 2733 . . . . . 6 (+gβ€˜πΉ) = (+gβ€˜πΉ)
1410, 11, 2, 12, 5, 13lmodvsdir 20361 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑂 ∈ (Baseβ€˜πΉ) ∧ 𝑂 ∈ (Baseβ€˜πΉ) ∧ 𝑋 ∈ 𝑉)) β†’ ((𝑂(+gβ€˜πΉ)𝑂) Β· 𝑋) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)))
151, 8, 8, 9, 14syl13anc 1373 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((𝑂(+gβ€˜πΉ)𝑂) Β· 𝑋) = ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)))
16 ringgrp 19974 . . . . . . 7 (𝐹 ∈ Ring β†’ 𝐹 ∈ Grp)
174, 16syl 17 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝐹 ∈ Grp)
185, 13, 6grplid 18785 . . . . . 6 ((𝐹 ∈ Grp ∧ 𝑂 ∈ (Baseβ€˜πΉ)) β†’ (𝑂(+gβ€˜πΉ)𝑂) = 𝑂)
1917, 8, 18syl2anc 585 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂(+gβ€˜πΉ)𝑂) = 𝑂)
2019oveq1d 7373 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((𝑂(+gβ€˜πΉ)𝑂) Β· 𝑋) = (𝑂 Β· 𝑋))
2115, 20eqtr3d 2775 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)) = (𝑂 Β· 𝑋))
2210, 2, 12, 5lmodvscl 20354 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑂 ∈ (Baseβ€˜πΉ) ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) ∈ 𝑉)
231, 8, 9, 22syl3anc 1372 . . . 4 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) ∈ 𝑉)
24 lmod0vs.z . . . . 5 0 = (0gβ€˜π‘Š)
2510, 11, 24lmod0vid 20369 . . . 4 ((π‘Š ∈ LMod ∧ (𝑂 Β· 𝑋) ∈ 𝑉) β†’ (((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)) = (𝑂 Β· 𝑋) ↔ 0 = (𝑂 Β· 𝑋)))
2623, 25syldan 592 . . 3 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (((𝑂 Β· 𝑋)(+gβ€˜π‘Š)(𝑂 Β· 𝑋)) = (𝑂 Β· 𝑋) ↔ 0 = (𝑂 Β· 𝑋)))
2721, 26mpbid 231 . 2 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 0 = (𝑂 Β· 𝑋))
2827eqcomd 2739 1 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (𝑂 Β· 𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  0gc0g 17326  Grpcgrp 18753  Ringcrg 19969  LModclmod 20336
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-ring 19971  df-lmod 20338
This theorem is referenced by:  lmodvs0  20371  lmodvsmmulgdi  20372  lcomfsupp  20377  lmodvneg1  20380  mptscmfsupp0  20402  lvecvs0or  20585  lssvs0or  20587  lspsneleq  20592  lspdisj  20602  lspfixed  20605  lspexch  20606  lspsolvlem  20619  lspsolv  20620  uvcresum  21215  frlmsslsp  21218  frlmup1  21220  frlmup2  21221  ascl0  21303  mplcoe1  21454  mplbas2  21459  ply10s0  21643  ply1scl0  21677  gsummoncoe1  21691  pmatcollpwscmatlem1  22154  idpm2idmp  22166  mp2pm2mplem4  22174  pm2mpmhmlem1  22183  monmat2matmon  22189  cpmidpmatlem3  22237  clm0vs  24474  plypf1  25589  lmodslmd  32088  evls1fpws  32320  ply1degltdimlem  32374  lbsdiflsp0  32378  fedgmullem2  32382  lshpkrlem1  37618  ldual0vs  37668  lclkrlem1  40015  lcd0vs  40124  baerlem3lem1  40216  baerlem5blem1  40218  hdmap14lem2a  40376  hdmap14lem4a  40380  hdmap14lem6  40382  hgmapval0  40401  prjspersym  40988  prjspreln0  40990  prjspner1  41007  lmod0rng  46252  scmsuppss  46534  lmodvsmdi  46544  ply1mulgsumlem4  46556  lincval1  46586  lincvalsc0  46588  linc0scn0  46590  linc1  46592  ldepsprlem  46639
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