Theorem lmodvsghm 21018

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 2769	. 2 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		lmodgrp 20962	. . 3 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
4	3	adantr 485	. 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 20973	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
9	8	3expa 1134	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝑅 · 𝑥) ∈ 𝑉)
10	9	fmpttd 7108	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)):𝑉⟶𝑉)
11	1, 2, 5, 6, 7	lmodvsdi 20980	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
12	11	3exp2 1371	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑅 ∈ 𝐾 → (𝑦 ∈ 𝑉 → (𝑧 ∈ 𝑉 → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧))))))
13	12	imp43 432	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑅 · (𝑦(+g‘𝑊)𝑧)) = ((𝑅 · 𝑦)(+g‘𝑊)(𝑅 · 𝑧)))
14	1, 2	lmodvacl 20970	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
15	14	3expb 1136	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
16	15	adantlr 727	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑉)
17		oveq2 7416	. . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
18		eqid 2769	. . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) = (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))
19		ovex 7441	. . . . 5 ⊢ (𝑅 · (𝑦(+g‘𝑊)𝑧)) ∈ V
20	17, 18, 19	fvmpt 6987	. . . 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 7416	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23		ovex 7441	. . . . . 6 ⊢ (𝑅 · 𝑦) ∈ V
24	22, 18, 23	fvmpt 6987	. . . . 5 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25		oveq2 7416	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26		ovex 7441	. . . . . 6 ⊢ (𝑅 · 𝑧) ∈ V
27	25, 18, 26	fvmpt 6987	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
28	24, 27	oveqan12d 7427	. . . 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 486	. . 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 2814	. 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 19291	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5193  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  +_gcplusg 17306  Scalarcsca 17309   ·_𝑠 cvsca 17310  Grpcgrp 18996   GrpHom cghm 19279  LModclmod 20955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-map 8822  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-grp 18999  df-ghm 19280  df-lmod 20957

This theorem is referenced by:  gsumvsmul  21021  lmhmvsca  21140