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Theorem lmodvsghm 19148
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 2817 . 2 (+g𝑊) = (+g𝑊)
3 lmodgrp 19094 . . 3 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
43adantr 468 . 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 19104 . . . 4 ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
983expa 1140 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ 𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
109fmpttd 6617 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)):𝑉𝑉)
111, 2, 5, 6, 7lmodvsdi 19110 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
12113exp2 1456 . . . 4 (𝑊 ∈ LMod → (𝑅𝐾 → (𝑦𝑉 → (𝑧𝑉 → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧))))))
1312imp43 416 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
141, 2lmodvacl 19101 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑦𝑉𝑧𝑉) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
15143expb 1142 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
1615adantlr 697 . . . 4 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
17 oveq2 6892 . . . . 5 (𝑥 = (𝑦(+g𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g𝑊)𝑧)))
18 eqid 2817 . . . . 5 (𝑥𝑉 ↦ (𝑅 · 𝑥)) = (𝑥𝑉 ↦ (𝑅 · 𝑥))
19 ovex 6916 . . . . 5 (𝑅 · (𝑦(+g𝑊)𝑧)) ∈ V
2017, 18, 19fvmpt 6513 . . . 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 6892 . . . . . 6 (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23 ovex 6916 . . . . . 6 (𝑅 · 𝑦) ∈ V
2422, 18, 23fvmpt 6513 . . . . 5 (𝑦𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25 oveq2 6892 . . . . . 6 (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26 ovex 6916 . . . . . 6 (𝑅 · 𝑧) ∈ V
2725, 18, 26fvmpt 6513 . . . . 5 (𝑧𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
2824, 27oveqan12d 6903 . . . 4 ((𝑦𝑉𝑧𝑉) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
2928adantl 469 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
3013, 21, 293eqtr4d 2861 . 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 17891 1 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  cmpt 4934  cfv 6111  (class class class)co 6884  Basecbs 16088  +gcplusg 16173  Scalarcsca 16176   ·𝑠 cvsca 16177  Grpcgrp 17647   GrpHom cghm 17879  LModclmod 19087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-mgm 17467  df-sgrp 17509  df-mnd 17520  df-grp 17650  df-ghm 17880  df-lmod 19089
This theorem is referenced by:  gsumvsmul  19151  lmhmvsca  19272
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