Theorem lmodvsghm 21094

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 2765	. 2 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		lmodgrp 21038	. . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
4	3	adantr 486	. 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 21049	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
9	8	3expa 1136	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
10	9	fmpttd 7114	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)):𝑉⟶𝑉)
11	1, 2, 5, 6, 7	lmodvsdi 21056	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
12	11	3exp2 1373	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧))))))
13	12	imp43 433	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
14	1, 2	lmodvacl 21046	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
15	14	3expb 1138	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
16	15	adantlr 728	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
17		oveq2 7427	. . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
18		eqid 2765	. . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) = (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))
19		ovex 7452	. . . . 5 ⊢ (𝑅 · (𝑦(+g‘𝑊)𝑧)) ∈ V
20	17, 18, 19	fvmpt 6993	. . . 4 ⊢ ((𝑦(+g‘𝑊)𝑧) ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g‘𝑊)𝑧)) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
21	16, 20	syl 18	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘(𝑦(+g‘𝑊)𝑧)) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
22		oveq2 7427	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23		ovex 7452	. . . . . 6 ⊢ (𝑅 · 𝑦) ∈ V
24	22, 18, 23	fvmpt 6993	. . . . 5 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25		oveq2 7427	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26		ovex 7452	. . . . . 6 ⊢ (𝑅 · 𝑧) ∈ V
27	25, 18, 26	fvmpt 6993	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
28	24, 27	oveqan12d 7438	. . . 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 487	. . 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 2810	. 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 19339	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  Scalarcsca 17335   ·_𝑠 cvsca 17336  Grpcgrp 19044   GrpHom cghm 19327  LModclmod 21031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-grp 19047  df-ghm 19328  df-lmod 21033

This theorem is used by:  gsumvsmul  21097  lmhmvsca  21216