Theorem isghm 13521

Description: Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)

Hypotheses

Ref	Expression
isghm.w	⊢ 𝑋 = (Base‘𝑆)
isghm.x	⊢ 𝑌 = (Base‘𝑇)
isghm.a	⊢ + = (+g‘𝑆)
isghm.b	⊢ ⨣ = (+g‘𝑇)

Assertion

Ref	Expression
isghm	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Distinct variable groups:   𝑣,𝑢,𝑆   𝑢,𝑇,𝑣   𝑢,𝑋,𝑣   𝑢, + ,𝑣   𝑢,𝑌,𝑣   𝑢, ⨣ ,𝑣   𝑢,𝐹,𝑣

Proof of Theorem isghm

Dummy variables 𝑡 𝑠 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ghm 13519	. . 3 ⊢ GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑓 ∣ [(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))})
2	1	elmpocl 6140	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3		isghm.w	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑆)
4		basfn 12832	. . . . . . . . 9 ⊢ Base Fn V
5		elex 2782	. . . . . . . . . 10 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ V)
6	5	adantr 276	. . . . . . . . 9 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝑆 ∈ V)
7		funfvex 5592	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
8	7	funfni 5375	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
9	4, 6, 8	sylancr 414	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (Base‘𝑆) ∈ V)
10	3, 9	eqeltrid 2291	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝑋 ∈ V)
11		isghm.x	. . . . . . . 8 ⊢ 𝑌 = (Base‘𝑇)
12		elex 2782	. . . . . . . . . 10 ⊢ (𝑇 ∈ Grp → 𝑇 ∈ V)
13	12	adantl 277	. . . . . . . . 9 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝑇 ∈ V)
14		funfvex 5592	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝑇 ∈ dom Base) → (Base‘𝑇) ∈ V)
15	14	funfni 5375	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝑇 ∈ V) → (Base‘𝑇) ∈ V)
16	4, 13, 15	sylancr 414	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (Base‘𝑇) ∈ V)
17	11, 16	eqeltrid 2291	. . . . . . 7 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝑌 ∈ V)
18		mapex 6740	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V)
19	10, 17, 18	syl2anc 411	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V)
20		simpl 109	. . . . . . . 8 ⊢ ((𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) → 𝑓:𝑋⟶𝑌)
21	20	ss2abi 3264	. . . . . . 7 ⊢ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ⊆ {𝑓 ∣ 𝑓:𝑋⟶𝑌}
22	21	a1i 9	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ⊆ {𝑓 ∣ 𝑓:𝑋⟶𝑌})
23	19, 22	ssexd 4183	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ∈ V)
24		vex 2774	. . . . . . . . . 10 ⊢ 𝑠 ∈ V
25		funfvex 5592	. . . . . . . . . . 11 ⊢ ((Fun Base ∧ 𝑠 ∈ dom Base) → (Base‘𝑠) ∈ V)
26	25	funfni 5375	. . . . . . . . . 10 ⊢ ((Base Fn V ∧ 𝑠 ∈ V) → (Base‘𝑠) ∈ V)
27	4, 24, 26	mp2an 426	. . . . . . . . 9 ⊢ (Base‘𝑠) ∈ V
28		feq2 5408	. . . . . . . . . 10 ⊢ (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
29		raleq 2701	. . . . . . . . . . 11 ⊢ (𝑤 = (Base‘𝑠) → (∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
30	29	raleqbi1dv 2713	. . . . . . . . . 10 ⊢ (𝑤 = (Base‘𝑠) → (∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
31	28, 30	anbi12d 473	. . . . . . . . 9 ⊢ (𝑤 = (Base‘𝑠) → ((𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))))
32	27, 31	sbcie 3032	. . . . . . . 8 ⊢ ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
33		fveq2 5575	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
34	33, 3	eqtr4di 2255	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
35	34	feq2d 5412	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶(Base‘𝑡)))
36		fveq2 5575	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
37		isghm.a	. . . . . . . . . . . . . 14 ⊢ + = (+g‘𝑆)
38	36, 37	eqtr4di 2255	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
39	38	oveqd 5960	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (𝑢(+g‘𝑠)𝑣) = (𝑢 + 𝑣))
40	39	fveqeq2d 5583	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → ((𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
41	34, 40	raleqbidv 2717	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
42	34, 41	raleqbidv 2717	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
43	35, 42	anbi12d 473	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))))
44	32, 43	bitrid 192	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))))
45	44	abbidv 2322	. . . . . 6 ⊢ (𝑠 = 𝑆 → {𝑓 ∣ [(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))})
46		fveq2 5575	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
47	46, 11	eqtr4di 2255	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
48	47	feq3d 5413	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑓:𝑋⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶𝑌))
49		fveq2 5575	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
50		isghm.b	. . . . . . . . . . . 12 ⊢ ⨣ = (+g‘𝑇)
51	49, 50	eqtr4di 2255	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
52	51	oveqd 5960	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))
53	52	eqeq2d 2216	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))
54	53	2ralbidv 2529	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))
55	48, 54	anbi12d 473	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))))
56	55	abbidv 2322	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
57	45, 56, 1	ovmpog 6079	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ∈ V) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
58	23, 57	mpd3an3 1350	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
59	58	eleq2d 2274	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))}))
60		simpr 110	. . . . . . 7 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌)
61	10	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 ∈ V)
62	60, 61	fexd 5813	. . . . . 6 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝐹:𝑋⟶𝑌) → 𝐹 ∈ V)
63	62	ex 115	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹:𝑋⟶𝑌 → 𝐹 ∈ V))
64	63	adantrd 279	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) → 𝐹 ∈ V))
65		feq1 5407	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋⟶𝑌 ↔ 𝐹:𝑋⟶𝑌))
66		fveq1 5574	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
67		fveq1 5574	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑢) = (𝐹‘𝑢))
68		fveq1 5574	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑣) = (𝐹‘𝑣))
69	67, 68	oveq12d 5961	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))
70	66, 69	eqeq12d 2219	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))
71	70	2ralbidv 2529	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))
72	65, 71	anbi12d 473	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
73	72	elab3g 2923	. . . 4 ⊢ (((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) → 𝐹 ∈ V) → (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
74	64, 73	syl 14	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
75	59, 74	bitrd 188	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
76	2, 75	biadanii 613	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  Vcvv 2771  [wsbc 2997   ⊆ wss 3165   Fn wfn 5265  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943  Basecbs 12774  +_gcplusg 12851  Grpcgrp 13274   GrpHom cghm 13518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-inn 9036  df-ndx 12777  df-slot 12778  df-base 12780  df-ghm 13519

This theorem is referenced by:  isghm3  13522  ghmgrp1  13523  ghmgrp2  13524  ghmf  13525  ghmlin  13526  isghmd  13530  idghm  13537  ghmf1o  13553  rhmopp  13880  expghmap  14311  mulgghm2  14312