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Theorem ghmsub 17876
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 17870 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
213ad2ant1 1127 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑆 ∈ Grp)
3 simp3 1132 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
4 ghmsub.b . . . . . 6 𝐵 = (Base‘𝑆)
5 eqid 2771 . . . . . 6 (invg𝑆) = (invg𝑆)
64, 5grpinvcl 17675 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
72, 3, 6syl2anc 573 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
8 eqid 2771 . . . . 5 (+g𝑆) = (+g𝑆)
9 eqid 2771 . . . . 5 (+g𝑇) = (+g𝑇)
104, 8, 9ghmlin 17873 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵 ∧ ((invg𝑆)‘𝑉) ∈ 𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
117, 10syld3an3 1515 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
12 eqid 2771 . . . . . 6 (invg𝑇) = (invg𝑇)
134, 5, 12ghminv 17875 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
14133adant2 1125 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
1514oveq2d 6809 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
1611, 15eqtrd 2805 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
17 ghmsub.m . . . . 5 = (-g𝑆)
184, 8, 5, 17grpsubval 17673 . . . 4 ((𝑈𝐵𝑉𝐵) → (𝑈 𝑉) = (𝑈(+g𝑆)((invg𝑆)‘𝑉)))
1918fveq2d 6336 . . 3 ((𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
20193adant1 1124 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
21 eqid 2771 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
224, 21ghmf 17872 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
23 ffvelrn 6500 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑈𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
24 ffvelrn 6500 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
2523, 24anim12dan 605 . . . . 5 ((𝐹:𝐵⟶(Base‘𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
2622, 25sylan 569 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
27263impb 1107 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
28 ghmsub.n . . . 4 𝑁 = (-g𝑇)
2921, 9, 12, 28grpsubval 17673 . . 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 2815 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)𝑁(𝐹𝑉)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Grpcgrp 17630  invgcminusg 17631  -gcsg 17632   GrpHom cghm 17865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-ghm 17866
This theorem is referenced by:  ghmnsgima  17892  ghmnsgpreima  17893  ghmeqker  17895  ghmf1  17897  evl1subd  19921  ghmcnp  22138  nmods  22768  qqhucn  30376
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