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Theorem ghmsub 17608
 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 17602 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
213ad2ant1 1080 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑆 ∈ Grp)
3 simp3 1061 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
4 ghmsub.b . . . . . 6 𝐵 = (Base‘𝑆)
5 eqid 2621 . . . . . 6 (invg𝑆) = (invg𝑆)
64, 5grpinvcl 17407 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
72, 3, 6syl2anc 692 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
8 eqid 2621 . . . . 5 (+g𝑆) = (+g𝑆)
9 eqid 2621 . . . . 5 (+g𝑇) = (+g𝑇)
104, 8, 9ghmlin 17605 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵 ∧ ((invg𝑆)‘𝑉) ∈ 𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
117, 10syld3an3 1368 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
12 eqid 2621 . . . . . 6 (invg𝑇) = (invg𝑇)
134, 5, 12ghminv 17607 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
14133adant2 1078 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
1514oveq2d 6631 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
1611, 15eqtrd 2655 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
17 ghmsub.m . . . . 5 = (-g𝑆)
184, 8, 5, 17grpsubval 17405 . . . 4 ((𝑈𝐵𝑉𝐵) → (𝑈 𝑉) = (𝑈(+g𝑆)((invg𝑆)‘𝑉)))
1918fveq2d 6162 . . 3 ((𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
20193adant1 1077 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
21 eqid 2621 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
224, 21ghmf 17604 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
23 ffvelrn 6323 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑈𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
24 ffvelrn 6323 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
2523, 24anim12dan 881 . . . . 5 ((𝐹:𝐵⟶(Base‘𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
2622, 25sylan 488 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈𝐵𝑉𝐵)) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
27263impb 1257 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
28 ghmsub.n . . . 4 𝑁 = (-g𝑇)
2921, 9, 12, 28grpsubval 17405 . . 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 2665 1 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = ((𝐹𝑈)𝑁(𝐹𝑉)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ⟶wf 5853  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Grpcgrp 17362  invgcminusg 17363  -gcsg 17364   GrpHom cghm 17597 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-ghm 17598 This theorem is referenced by:  ghmnsgima  17624  ghmnsgpreima  17625  ghmeqker  17627  ghmf1  17629  evl1subd  19646  ghmcnp  21858  nmods  22488  qqhucn  29860
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