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Theorem lmodvsghm 20398
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 2733 . 2 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
3 lmodgrp 20343 . . 3 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
43adantr 482 . 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 20354 . . . 4 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ π‘₯ ∈ 𝑉) β†’ (𝑅 Β· π‘₯) ∈ 𝑉)
983expa 1119 . . 3 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ π‘₯ ∈ 𝑉) β†’ (𝑅 Β· π‘₯) ∈ 𝑉)
109fmpttd 7064 . 2 ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) β†’ (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)):π‘‰βŸΆπ‘‰)
111, 2, 5, 6, 7lmodvsdi 20360 . . . . 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 1355 . . . 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 429 . . 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 20351 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
15143expb 1121 . . . . 5 ((π‘Š ∈ LMod ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
1615adantlr 714 . . . 4 (((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾) ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
17 oveq2 7366 . . . . 5 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (𝑅 Β· π‘₯) = (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)))
18 eqid 2733 . . . . 5 (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯)) = (π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))
19 ovex 7391 . . . . 5 (𝑅 Β· (𝑦(+gβ€˜π‘Š)𝑧)) ∈ V
2017, 18, 19fvmpt 6949 . . . 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 7366 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑦))
23 ovex 7391 . . . . . 6 (𝑅 Β· 𝑦) ∈ V
2422, 18, 23fvmpt 6949 . . . . 5 (𝑦 ∈ 𝑉 β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘¦) = (𝑅 Β· 𝑦))
25 oveq2 7366 . . . . . 6 (π‘₯ = 𝑧 β†’ (𝑅 Β· π‘₯) = (𝑅 Β· 𝑧))
26 ovex 7391 . . . . . 6 (𝑅 Β· 𝑧) ∈ V
2725, 18, 26fvmpt 6949 . . . . 5 (𝑧 ∈ 𝑉 β†’ ((π‘₯ ∈ 𝑉 ↦ (𝑅 Β· π‘₯))β€˜π‘§) = (𝑅 Β· 𝑧))
2824, 27oveqan12d 7377 . . . 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 483 . . 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 2783 . 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 19022 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 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141   ·𝑠 cvsca 17142  Grpcgrp 18753   GrpHom cghm 19010  LModclmod 20336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-ghm 19011  df-lmod 20338
This theorem is referenced by:  gsumvsmul  20401  lmhmvsca  20521
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