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Theorem ghmsub 18107
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 18101 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
213ad2ant1 1126 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑆 ∈ Grp)
3 simp3 1131 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → 𝑉𝐵)
4 ghmsub.b . . . . . 6 𝐵 = (Base‘𝑆)
5 eqid 2795 . . . . . 6 (invg𝑆) = (invg𝑆)
64, 5grpinvcl 17908 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
72, 3, 6syl2anc 584 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((invg𝑆)‘𝑉) ∈ 𝐵)
8 eqid 2795 . . . . 5 (+g𝑆) = (+g𝑆)
9 eqid 2795 . . . . 5 (+g𝑇) = (+g𝑇)
104, 8, 9ghmlin 18104 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵 ∧ ((invg𝑆)‘𝑉) ∈ 𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
117, 10syld3an3 1402 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))))
12 eqid 2795 . . . . . 6 (invg𝑇) = (invg𝑇)
134, 5, 12ghminv 18106 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
14133adant2 1124 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘((invg𝑆)‘𝑉)) = ((invg𝑇)‘(𝐹𝑉)))
1514oveq2d 7032 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈)(+g𝑇)(𝐹‘((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
1611, 15eqtrd 2831 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))) = ((𝐹𝑈)(+g𝑇)((invg𝑇)‘(𝐹𝑉))))
17 ghmsub.m . . . . 5 = (-g𝑆)
184, 8, 5, 17grpsubval 17906 . . . 4 ((𝑈𝐵𝑉𝐵) → (𝑈 𝑉) = (𝑈(+g𝑆)((invg𝑆)‘𝑉)))
1918fveq2d 6542 . . 3 ((𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
20193adant1 1123 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → (𝐹‘(𝑈 𝑉)) = (𝐹‘(𝑈(+g𝑆)((invg𝑆)‘𝑉))))
21 eqid 2795 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
224, 21ghmf 18103 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝐵⟶(Base‘𝑇))
23 ffvelrn 6714 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑈𝐵) → (𝐹𝑈) ∈ (Base‘𝑇))
24 ffvelrn 6714 . . . . . 6 ((𝐹:𝐵⟶(Base‘𝑇) ∧ 𝑉𝐵) → (𝐹𝑉) ∈ (Base‘𝑇))
2523, 24anim12dan 618 . . . . 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 1108 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈𝐵𝑉𝐵) → ((𝐹𝑈) ∈ (Base‘𝑇) ∧ (𝐹𝑉) ∈ (Base‘𝑇)))
28 ghmsub.n . . . 4 𝑁 = (-g𝑇)
2921, 9, 12, 28grpsubval 17906 . . 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 2841 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 1080   = wceq 1522  wcel 2081  wf 6221  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  Grpcgrp 17861  invgcminusg 17862  -gcsg 17863   GrpHom cghm 18096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-0g 16544  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-grp 17864  df-minusg 17865  df-sbg 17866  df-ghm 18097
This theorem is referenced by:  ghmnsgima  18123  ghmnsgpreima  18124  ghmeqker  18126  ghmf1  18128  evl1subd  20187  ghmcnp  22406  nmods  23036  qqhucn  30850
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