Theorem lmodvsghm 20938

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	⊢ · = ( ·𝑠 ‘𝑊)
lmodvsghm.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
lmodvsghm	⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (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.

Step	Hyp	Ref	Expression
1		lmodvsghm.v	. 2 ⊢ 𝑉 = (Base‘𝑊)
2		eqid 2735	. 2 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		lmodgrp 20882	. . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
4	3	adantr 480	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → 𝑊 ∈ Grp)
5		lmodvsghm.f	. . . . 5 ⊢ 𝐹 = (Scalar‘𝑊)
6		lmodvsghm.s	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		lmodvsghm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐹)
8	1, 5, 6, 7	lmodvscl 20893	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
9	8	3expa 1117	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
10	9	fmpttd 7135	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)):𝑉⟶𝑉)
11	1, 2, 5, 6, 7	lmodvsdi 20900	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
12	11	3exp2 1353	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧))))))
13	12	imp43 427	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
14	1, 2	lmodvacl 20890	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
15	14	3expb 1119	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
16	15	adantlr 715	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
17		oveq2 7439	. . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
18		eqid 2735	. . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) = (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))
19		ovex 7464	. . . . 5 ⊢ (𝑅 · (𝑦(+g‘𝑊)𝑧)) ∈ V
20	17, 18, 19	fvmpt 7016	. . . 4 ⊢ ((𝑦(+g‘𝑊)𝑧) ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g‘𝑊)𝑧)) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
21	16, 20	syl 17	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g‘𝑊)𝑧)) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
22		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23		ovex 7464	. . . . . 6 ⊢ (𝑅 · 𝑦) ∈ V
24	22, 18, 23	fvmpt 7016	. . . . 5 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26		ovex 7464	. . . . . 6 ⊢ (𝑅 · 𝑧) ∈ V
27	25, 18, 26	fvmpt 7016	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
28	24, 27	oveqan12d 7450	. . . 4 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g‘𝑊)((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
29	28	adantl 481	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g‘𝑊)((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
30	13, 21, 29	3eqtr4d 2785	. 2 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g‘𝑊)𝑧)) = (((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦)(+g‘𝑊)((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧)))
31	1, 1, 2, 2, 4, 4, 10, 30	isghmd 19256	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Scalarcsca 17301   ·_𝑠 cvsca 17302  Grpcgrp 18964   GrpHom cghm 19243  LModclmod 20875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ghm 19244  df-lmod 20877

This theorem is referenced by:  gsumvsmul  20941  lmhmvsca  21062