Theorem lmodvsghm 21207

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 2760	. 2 ⊢ (+g‘𝑊) = (+g‘𝑊)
3		lmodgrp 21151	. . 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 21162	. . . 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 7120	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)):𝑉⟶𝑉)
11	1, 2, 5, 6, 7	lmodvsdi 21169	. . . . 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 21159	. . . . . 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 7432	. . . . 5 ⊢ (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑅 · 𝑥) = (𝑅 · (𝑦(+g‘𝑊)𝑧)))
18		eqid 2760	. . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) = (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))
19		ovex 7457	. . . . 5 ⊢ (𝑅 · (𝑦(+g‘𝑊)𝑧)) ∈ V
20	17, 18, 19	fvmpt 6998	. . . 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 7432	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑅 · 𝑥) = (𝑅 · 𝑦))
23		ovex 7457	. . . . . 6 ⊢ (𝑅 · 𝑦) ∈ V
24	22, 18, 23	fvmpt 6998	. . . . 5 ⊢ (𝑦 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑦) = (𝑅 · 𝑦))
25		oveq2 7432	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑅 · 𝑥) = (𝑅 · 𝑧))
26		ovex 7457	. . . . . 6 ⊢ (𝑅 · 𝑧) ∈ V
27	25, 18, 26	fvmpt 6998	. . . . 5 ⊢ (𝑧 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥))‘𝑧) = (𝑅 · 𝑧))
28	24, 27	oveqan12d 7443	. . . 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 2805	. 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 19448	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  +_gcplusg 17437  Scalarcsca 17440   ·_𝑠 cvsca 17441  Grpcgrp 19153   GrpHom cghm 19436  LModclmod 21144

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8856  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-ghm 19437  df-lmod 21146

This theorem is used by:  gsumvsmul  21210  lmhmvsca  21329