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Theorem lmod0vs 20505
Description: Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 30263 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 20479 . . . . . . 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 20084 . . . . . 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 20496 . . . . 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 20061 . . . . . . 7 (𝐹 ∈ Ring β†’ 𝐹 ∈ Grp)
174, 16syl 17 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝐹 ∈ Grp)
185, 13, 6grplid 18852 . . . . . 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 7424 . . . 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 20489 . . . . 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 20504 . . . 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 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  Grpcgrp 18819  Ringcrg 20056  LModclmod 20471
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-ring 20058  df-lmod 20473
This theorem is referenced by:  lmodvs0  20506  lmodvsmmulgdi  20507  lcomfsupp  20512  lmodvneg1  20515  mptscmfsupp0  20537  lvecvs0or  20721  lssvs0or  20723  lspsneleq  20728  lspdisj  20738  lspfixed  20741  lspexch  20742  lspsolvlem  20755  lspsolv  20756  uvcresum  21348  frlmsslsp  21351  frlmup1  21353  frlmup2  21354  ascl0  21438  mplcoe1  21592  mplbas2  21597  ply10s0  21778  ply1scl0OLD  21813  gsummoncoe1  21828  pmatcollpwscmatlem1  22291  idpm2idmp  22303  mp2pm2mplem4  22311  pm2mpmhmlem1  22320  monmat2matmon  22326  cpmidpmatlem3  22374  clm0vs  24611  plypf1  25726  lmodslmd  32349  evls1fpws  32646  ply1degltdimlem  32707  lbsdiflsp0  32711  fedgmullem2  32715  lshpkrlem1  37980  ldual0vs  38030  lclkrlem1  40377  lcd0vs  40486  baerlem3lem1  40578  baerlem5blem1  40580  hdmap14lem2a  40738  hdmap14lem4a  40742  hdmap14lem6  40744  hgmapval0  40763  selvvvval  41157  prjspersym  41349  prjspreln0  41351  prjspner1  41368  lmod0rng  46642  scmsuppss  47048  lmodvsmdi  47058  ply1mulgsumlem4  47070  lincval1  47100  lincvalsc0  47102  linc0scn0  47104  linc1  47106  ldepsprlem  47153
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