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Theorem lmodvsghm 20921
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 2737 . 2 (+g𝑊) = (+g𝑊)
3 lmodgrp 20865 . . 3 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
43adantr 480 . 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 20876 . . . 4 ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
983expa 1119 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ 𝑥𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
109fmpttd 7135 . 2 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)):𝑉𝑉)
111, 2, 5, 6, 7lmodvsdi 20883 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
12113exp2 1355 . . . 4 (𝑊 ∈ LMod → (𝑅𝐾 → (𝑦𝑉 → (𝑧𝑉 → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧))))))
1312imp43 427 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑅 · (𝑦(+g𝑊)𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
141, 2lmodvacl 20873 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑦𝑉𝑧𝑉) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
15143expb 1121 . . . . 5 ((𝑊 ∈ LMod ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
1615adantlr 715 . . . 4 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (𝑦(+g𝑊)𝑧) ∈ 𝑉)
17 oveq2 7439 . . . . 5 (𝑥 = (𝑦(+g𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g𝑊)𝑧)))
18 eqid 2737 . . . . 5 (𝑥𝑉 ↦ (𝑅 · 𝑥)) = (𝑥𝑉 ↦ (𝑅 · 𝑥))
19 ovex 7464 . . . . 5 (𝑅 · (𝑦(+g𝑊)𝑧)) ∈ V
2017, 18, 19fvmpt 7016 . . . 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 7439 . . . . . 6 (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23 ovex 7464 . . . . . 6 (𝑅 · 𝑦) ∈ V
2422, 18, 23fvmpt 7016 . . . . 5 (𝑦𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25 oveq2 7439 . . . . . 6 (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26 ovex 7464 . . . . . 6 (𝑅 · 𝑧) ∈ V
2725, 18, 26fvmpt 7016 . . . . 5 (𝑧𝑉 → ((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
2824, 27oveqan12d 7450 . . . 4 ((𝑦𝑉𝑧𝑉) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
2928adantl 481 . . 3 (((𝑊 ∈ LMod ∧ 𝑅𝐾) ∧ (𝑦𝑉𝑧𝑉)) → (((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g𝑊)((𝑥𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g𝑊)(𝑅 · 𝑧)))
3013, 21, 293eqtr4d 2787 . 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 19243 1 ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  Grpcgrp 18951   GrpHom cghm 19230  LModclmod 20858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ghm 19231  df-lmod 20860
This theorem is referenced by:  gsumvsmul  20924  lmhmvsca  21044
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