Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmodvsghm Structured version   Visualization version   GIF version

Theorem lmodvsghm 19691
 Description: Scalar multiplication of the vector space by a fixed scalar is an endomorphism of the additive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
Hypotheses
Ref Expression
lmodvsghm.v 𝑉 = (Base‘𝑊)
lmodvsghm.f 𝐹 = (Scalar‘𝑊)
lmodvsghm.s · = ( ·𝑠𝑊)
lmodvsghm.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
lmodvsghm ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑅   𝑥, ·   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem lmodvsghm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodvsghm.v . 2 𝑉 = (Base‘𝑊)
2 eqid 2798 . 2 (+g𝑊) = (+g𝑊)
3 lmodgrp 19637 . . 3 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
43adantr 484 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → 𝑊 ∈ Grp)
5 lmodvsghm.f . . . . 5 𝐹 = (Scalar‘𝑊)
6 lmodvsghm.s . . . . 5 · = ( ·𝑠𝑊)
7 lmodvsghm.k . . . . 5 𝐾 = (Base‘𝐹)
81, 5, 6, 7lmodvscl 19647 . . . 4 ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
983expa 1115 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ 𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
109fmpttd 6856 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)):𝑉𝑉)
111, 2, 5, 6, 7lmodvsdi 19653 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
12113exp2 1351 . . . 4 (𝑊 ∈ LMod → (𝑅𝐾 → (𝑦𝑉 → (𝑧𝑉 → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧))))))
1312imp43 431 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
141, 2lmodvacl 19644 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑦𝑉𝑧𝑉) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
15143expb 1117 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
1615adantlr 714 . . . 4 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
17 oveq2 7143 . . . . 5 (𝑥 = (𝑦(+g𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g𝑊)𝑧)))
18 eqid 2798 . . . . 5 (𝑥𝑉 ↦ (𝑅 · 𝑥)) = (𝑥𝑉 ↦ (𝑅 · 𝑥))
19 ovex 7168 . . . . 5 (𝑅 · (𝑦(+g𝑊)𝑧)) ∈ V
2017, 18, 19fvmpt 6745 . . . 4 ((𝑦(+g𝑊)𝑧) ∈ 𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g𝑊)𝑧)) = (𝑅 · (𝑦(+g𝑊)𝑧)))
2116, 20syl 17 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g𝑊)𝑧)) = (𝑅 · (𝑦(+g𝑊)𝑧)))
22 oveq2 7143 . . . . . 6 (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23 ovex 7168 . . . . . 6 (𝑅 · 𝑦) ∈ V
2422, 18, 23fvmpt 6745 . . . . 5 (𝑦𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25 oveq2 7143 . . . . . 6 (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26 ovex 7168 . . . . . 6 (𝑅 · 𝑧) ∈ V
2725, 18, 26fvmpt 6745 . . . . 5 (𝑧𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
2824, 27oveqan12d 7154 . . . 4 ((𝑦𝑉𝑧𝑉) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
2928adantl 485 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
3013, 21, 293eqtr4d 2843 . 2 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g𝑊)𝑧)) = (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)))
311, 1, 2, 2, 4, 4, 10, 30isghmd 18362 1 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  +gcplusg 16559  Scalarcsca 16562   ·𝑠 cvsca 16563  Grpcgrp 18097   GrpHom cghm 18350  LModclmod 19630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-ghm 18351  df-lmod 19632 This theorem is referenced by:  gsumvsmul  19694  lmhmvsca  19813
 Copyright terms: Public domain W3C validator