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Theorem ghmlin 19252
Description: A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmlin.x 𝑋 = (Base‘𝑆)
ghmlin.a + = (+g𝑆)
ghmlin.b = (+g𝑇)
Assertion
Ref Expression
ghmlin ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉)))

Proof of Theorem ghmlin
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmlin.x . . . . . 6 𝑋 = (Base‘𝑆)
2 eqid 2735 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
3 ghmlin.a . . . . . 6 + = (+g𝑆)
4 ghmlin.b . . . . . 6 = (+g𝑇)
51, 2, 3, 4isghm 19246 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎𝑋𝑏𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)))))
65simprbi 496 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎𝑋𝑏𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏))))
76simprd 495 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎𝑋𝑏𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)))
8 fvoveq1 7454 . . . . 5 (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏)))
9 fveq2 6907 . . . . . 6 (𝑎 = 𝑈 → (𝐹𝑎) = (𝐹𝑈))
109oveq1d 7446 . . . . 5 (𝑎 = 𝑈 → ((𝐹𝑎) (𝐹𝑏)) = ((𝐹𝑈) (𝐹𝑏)))
118, 10eqeq12d 2751 . . . 4 (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹𝑈) (𝐹𝑏))))
12 oveq2 7439 . . . . . 6 (𝑏 = 𝑉 → (𝑈 + 𝑏) = (𝑈 + 𝑉))
1312fveq2d 6911 . . . . 5 (𝑏 = 𝑉 → (𝐹‘(𝑈 + 𝑏)) = (𝐹‘(𝑈 + 𝑉)))
14 fveq2 6907 . . . . . 6 (𝑏 = 𝑉 → (𝐹𝑏) = (𝐹𝑉))
1514oveq2d 7447 . . . . 5 (𝑏 = 𝑉 → ((𝐹𝑈) (𝐹𝑏)) = ((𝐹𝑈) (𝐹𝑉)))
1613, 15eqeq12d 2751 . . . 4 (𝑏 = 𝑉 → ((𝐹‘(𝑈 + 𝑏)) = ((𝐹𝑈) (𝐹𝑏)) ↔ (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉))))
1711, 16rspc2v 3633 . . 3 ((𝑈𝑋𝑉𝑋) → (∀𝑎𝑋𝑏𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹𝑎) (𝐹𝑏)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉))))
187, 17mpan9 506 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈𝑋𝑉𝑋)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉)))
19183impb 1114 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝑋𝑉𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹𝑈) (𝐹𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ghm 19244
This theorem is referenced by:  ghmid  19253  ghminv  19254  ghmsub  19255  ghmmhm  19257  ghmrn  19260  resghm  19263  ghmpreima  19269  ghmnsgima  19271  ghmnsgpreima  19272  ghmf1o  19279  ghmqusnsglem1  19311  ghmqusnsg  19313  ghmquskerlem1  19314  ghmquskerlem3  19317  lactghmga  19438  invghm  19866  ghmplusg  19879  rhmopp  20526  srngadd  20869  islmhm2  21055  rhmpreimaidl  21305  cygznlem3  21606  psgnco  21619  evpmodpmf1o  21632  ipdir  21675  evlslem1  22124  mpfind  22149  evl1addd  22361  mdetralt  22630  cpmatacl  22738  mat2pmatghm  22752  ghmcnp  24139  ply1rem  26220  dchrptlem2  27324  abliso  33024  rhmimaidl  33440  r1pquslmic  33611  dimkerim  33655  zrhcntr  33942  qqhghm  33951  qqhrhm  33952  fldhmf1  42072  aks6d1c1p3  42092  aks6d1c5lem1  42118  aks6d1c5lem2  42120  aks5lem3a  42171  evlsaddval  42555  evladdval  42562  gicabl  43088
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