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Theorem lmodvsghm 19133
 Description: Scalar multiplication of the vector space by a fixed scalar is an automorphism 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 2770 . 2 (+g𝑊) = (+g𝑊)
3 lmodgrp 19079 . . 3 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
43adantr 466 . 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 19089 . . . 4 ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
983expa 1110 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ 𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
10 eqid 2770 . . 3 (𝑥𝑉 ↦ (𝑅 · 𝑥)) = (𝑥𝑉 ↦ (𝑅 · 𝑥))
119, 10fmptd 6527 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)):𝑉𝑉)
121, 2, 5, 6, 7lmodvsdi 19095 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
13123exp2 1446 . . . 4 (𝑊 ∈ LMod → (𝑅𝐾 → (𝑦𝑉 → (𝑧𝑉 → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧))))))
1413imp43 414 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
151, 2lmodvacl 19086 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑦𝑉𝑧𝑉) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
16153expb 1112 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
1716adantlr 686 . . . 4 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
18 oveq2 6800 . . . . 5 (𝑥 = (𝑦(+g𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g𝑊)𝑧)))
19 ovex 6822 . . . . 5 (𝑅 · (𝑦(+g𝑊)𝑧)) ∈ V
2018, 10, 19fvmpt 6424 . . . 4 ((𝑦(+g𝑊)𝑧) ∈ 𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g𝑊)𝑧)) = (𝑅 · (𝑦(+g𝑊)𝑧)))
2117, 20syl 17 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g𝑊)𝑧)) = (𝑅 · (𝑦(+g𝑊)𝑧)))
22 oveq2 6800 . . . . . 6 (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23 ovex 6822 . . . . . 6 (𝑅 · 𝑦) ∈ V
2422, 10, 23fvmpt 6424 . . . . 5 (𝑦𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25 oveq2 6800 . . . . . 6 (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26 ovex 6822 . . . . . 6 (𝑅 · 𝑧) ∈ V
2725, 10, 26fvmpt 6424 . . . . 5 (𝑧𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
2824, 27oveqan12d 6811 . . . 4 ((𝑦𝑉𝑧𝑉) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
2928adantl 467 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
3014, 21, 293eqtr4d 2814 . 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, 11, 30isghmd 17876 1 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144   ↦ cmpt 4861  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151   ·𝑠 cvsca 16152  Grpcgrp 17629   GrpHom cghm 17864  LModclmod 19072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-ghm 17865  df-lmod 19074 This theorem is referenced by:  gsumvsmul  19136  lmhmvsca  19257
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