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Theorem ghmsub 19099
Description: Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmsub.b 𝐵 = (Base‘𝑆)
ghmsub.m = (-g𝑆)
ghmsub.n 𝑁 = (-g𝑇)
Assertion
Ref Expression
ghmsub ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)𝑁(𝐹𝑉)))

Proof of Theorem ghmsub
StepHypRef Expression
1 ghmgrp1 19093 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
213ad2ant1 1133 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑆 ∈ Grp)
3 simp3 1138 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
4 ghmsub.b . . . . . 6 𝐵 = (Base‘𝑆)
5 eqid 2732 . . . . . 6 (invg𝑆) = (invg𝑆)
64, 5grpinvcl 18871 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
72, 3, 6syl2anc 584 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
8 eqid 2732 . . . . 5 (+g𝑆) = (+g𝑆)
9 eqid 2732 . . . . 5 (+g𝑇) = (+g𝑇)
104, 8, 9ghmlin 19096 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵 ∧ ((invg𝑆)‘𝑉) ∈ 𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
117, 10syld3an3 1409 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
12 eqid 2732 . . . . . 6 (invg𝑇) = (invg𝑇)
134, 5, 12ghminv 19098 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
14133adant2 1131 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
1514oveq2d 7424 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
1611, 15eqtrd 2772 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
17 ghmsub.m . . . . 5 = (-g𝑆)
184, 8, 5, 17grpsubval 18869 . . . 4 ((𝑈𝐵𝑉𝐵) → (𝑈 𝑉) = (𝑈(+g𝑆)((invg𝑆)‘𝑉)))
1918fveq2d 6895 . . 3 ((𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
20193adant1 1130 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
21 eqid 2732 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
224, 21ghmf 19095 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
23 ffvelcdm 7083 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑈𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
24 ffvelcdm 7083 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
2523, 24anim12dan 619 . . . . 5 ((𝐹:𝐵⟶(Base‘𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
2622, 25sylan 580 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
27263impb 1115 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
28 ghmsub.n . . . 4 𝑁 = (-g𝑇)
2921, 9, 12, 28grpsubval 18869 . . 3 (((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)) → ((𝐹𝑈)𝑁(𝐹𝑉)) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
3027, 29syl 17 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈)𝑁(𝐹𝑉)) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
3116, 20, 303eqtr4d 2782 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)𝑁(𝐹𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wf 6539  cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  Grpcgrp 18818  invgcminusg 18819  -gcsg 18820   GrpHom cghm 19088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-ghm 19089
This theorem is referenced by:  ghmnsgima  19115  ghmnsgpreima  19116  ghmeqker  19118  ghmf1  19120  evl1subd  21860  ghmcnp  23618  nmods  24260  fermltlchr  32473  znfermltl  32474  qqhucn  32967
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