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Theorem lmodvsghm 20677
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 = (Baseβ€˜π‘Š)
lmodvsghm.f 𝐹 = (Scalarβ€˜π‘Š)
lmodvsghm.s Β· = ( ·𝑠 β€˜π‘Š)
lmodvsghm.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
lmodvsghm ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)) ∈ (π‘Š GrpHom π‘Š))
Distinct variable groups:   π‘₯,𝐾   π‘₯,𝑅   π‘₯, Β·   π‘₯,𝑉   π‘₯,π‘Š
Allowed substitution hint:   𝐹(π‘₯)
Proof of Theorem lmodvsghm
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmodvsghm.v . 2 𝑉 = (Baseβ€˜π‘Š)
2 eqid 2730 . 2 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
3 lmodgrp 20621 . . 3 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
43adantr 479 . 2 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ π‘Š ∈ Grp)
5 lmodvsghm.f . . . . 5 𝐹 = (Scalarβ€˜π‘Š)
6 lmodvsghm.s . . . . 5 Β· = ( ·𝑠 β€˜π‘Š)
7 lmodvsghm.k . . . . 5 𝐾 = (Baseβ€˜πΉ)
81, 5, 6, 7lmodvscl 20632 . . . 4 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ π‘₯ ∈ 𝑉) β†’ (𝑅 Β· π‘₯) ∈ 𝑉)
983expa 1116 . . 3 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ π‘₯ ∈ 𝑉) β†’ (𝑅 Β· π‘₯) ∈ 𝑉)
109fmpttd 7115 . 2 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)):π‘‰βŸΆπ‘‰)
111, 2, 5, 6, 7lmodvsdi 20639 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)) = ((𝑅 Β· 𝑦)(+gβ€˜π‘Š)(𝑅 Β· 𝑧)))
12113exp2 1352 . . . 4 (π‘Š ∈ LMod β†’ (𝑅 ∈ 𝐾 β†’ (𝑦 ∈ 𝑉 β†’ (𝑧 ∈ 𝑉 β†’ (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)) = ((𝑅 Β· 𝑦)(+gβ€˜π‘Š)(𝑅 Β· 𝑧))))))
1312imp43 426 . . 3 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)) = ((𝑅 Β· 𝑦)(+gβ€˜π‘Š)(𝑅 Β· 𝑧)))
141, 2lmodvacl 20629 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
15143expb 1118 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
1615adantlr 711 . . . 4 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
17 oveq2 7419 . . . . 5 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (𝑅 Β· π‘₯) = (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)))
18 eqid 2730 . . . . 5 (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)) = (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))
19 ovex 7444 . . . . 5 (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)) ∈ V
2017, 18, 19fvmpt 6997 . . . 4 ((𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉 β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜(𝑦(+gβ€˜π‘Š)𝑧)) = (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)))
2116, 20syl 17 . . 3 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜(𝑦(+gβ€˜π‘Š)𝑧)) = (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)))
22 oveq2 7419 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑦))
23 ovex 7444 . . . . . 6 (𝑅 Β· 𝑦) ∈ V
2422, 18, 23fvmpt 6997 . . . . 5 (𝑦 ∈ 𝑉 β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘¦) = (𝑅 Β· 𝑦))
25 oveq2 7419 . . . . . 6 (π‘₯ = 𝑧 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑧))
26 ovex 7444 . . . . . 6 (𝑅 Β· 𝑧) ∈ V
2725, 18, 26fvmpt 6997 . . . . 5 (𝑧 ∈ 𝑉 β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘§) = (𝑅 Β· 𝑧))
2824, 27oveqan12d 7430 . . . 4 ((𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ (((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘¦)(+gβ€˜π‘Š)((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘§)) = ((𝑅 Β· 𝑦)(+gβ€˜π‘Š)(𝑅 Β· 𝑧)))
2928adantl 480 . . 3 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘¦)(+gβ€˜π‘Š)((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘§)) = ((𝑅 Β· 𝑦)(+gβ€˜π‘Š)(𝑅 Β· 𝑧)))
3013, 21, 293eqtr4d 2780 . 2 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜(𝑦(+gβ€˜π‘Š)𝑧)) = (((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘¦)(+gβ€˜π‘Š)((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘§)))
311, 1, 2, 2, 4, 4, 10, 30isghmd 19139 1 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)) ∈ (π‘Š GrpHom π‘Š))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  Scalarcsca 17204   ·𝑠 cvsca 17205  Grpcgrp 18855   GrpHom cghm 19127  LModclmod 20614
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-ghm 19128  df-lmod 20616
This theorem is referenced by:  gsumvsmul  20680  lmhmvsca  20800
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