Theorem ghmlin 19150

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	⊢ + = (+g‘𝑆)
ghmlin.b	⊢ ⨣ = (+g‘𝑇)

Assertion

Ref	Expression
ghmlin	⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))

Proof of Theorem ghmlin

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmlin.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
2		eqid 2736	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		ghmlin.a	. . . . . 6 ⊢ + = (+g‘𝑆)
4		ghmlin.b	. . . . . 6 ⊢ ⨣ = (+g‘𝑇)
5	1, 2, 3, 4	isghm 19144	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)))))
6	5	simprbi 496	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋⟶(Base‘𝑇) ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏))))
7	6	simprd 495	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)))
8		fvoveq1 7381	. . . . 5 ⊢ (𝑎 = 𝑈 → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑈 + 𝑏)))
9		fveq2 6834	. . . . . 6 ⊢ (𝑎 = 𝑈 → (𝐹‘𝑎) = (𝐹‘𝑈))
10	9	oveq1d 7373	. . . . 5 ⊢ (𝑎 = 𝑈 → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)))
11	8, 10	eqeq12d 2752	. . . 4 ⊢ (𝑎 = 𝑈 → ((𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏))))
12		oveq2 7366	. . . . . 6 ⊢ (𝑏 = 𝑉 → (𝑈 + 𝑏) = (𝑈 + 𝑉))
13	12	fveq2d 6838	. . . . 5 ⊢ (𝑏 = 𝑉 → (𝐹‘(𝑈 + 𝑏)) = (𝐹‘(𝑈 + 𝑉)))
14		fveq2 6834	. . . . . 6 ⊢ (𝑏 = 𝑉 → (𝐹‘𝑏) = (𝐹‘𝑉))
15	14	oveq2d 7374	. . . . 5 ⊢ (𝑏 = 𝑉 → ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))
16	13, 15	eqeq12d 2752	. . . 4 ⊢ (𝑏 = 𝑉 → ((𝐹‘(𝑈 + 𝑏)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑏)) ↔ (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))))
17	11, 16	rspc2v 3587	. . 3 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉))))
18	7, 17	mpan9 506	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋)) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))
19	18	3impb 1114	1 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ∈ 𝑋) → (𝐹‘(𝑈 + 𝑉)) = ((𝐹‘𝑈) ⨣ (𝐹‘𝑉)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  Grpcgrp 18863   GrpHom cghm 19141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-ghm 19142

This theorem is referenced by:  ghmid  19151  ghminv  19152  ghmsub  19153  ghmmhm  19155  ghmrn  19158  resghm  19161  ghmpreima  19167  ghmnsgima  19169  ghmnsgpreima  19170  ghmf1o  19177  ghmqusnsglem1  19209  ghmqusnsg  19211  ghmquskerlem1  19212  ghmquskerlem3  19215  lactghmga  19334  invghm  19762  ghmplusg  19775  rhmopp  20442  srngadd  20784  islmhm2  20990  rhmpreimaidl  21232  cygznlem3  21524  psgnco  21538  evpmodpmf1o  21551  ipdir  21594  evlslem1  22037  evladdval  22058  mpfind  22070  evl1addd  22285  mdetralt  22552  cpmatacl  22660  mat2pmatghm  22674  ghmcnp  24059  ply1rem  26127  dchrptlem2  27232  abliso  33118  rhmimaidl  33513  r1pquslmic  33692  dimkerim  33784  zrhcntr  34136  qqhghm  34145  qqhrhm  34146  fldhmf1  42340  aks6d1c1p3  42360  aks6d1c5lem1  42386  aks6d1c5lem2  42388  aks5lem3a  42439  evlsaddval  42810  gicabl  43337