Theorem isghm 18930

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 18928	. . 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 7573	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3		fvex 6838	. . . . . . . 8 ⊢ (Base‘𝑠) ∈ V
4		feq2 6633	. . . . . . . . 9 ⊢ (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
5		raleq 3305	. . . . . . . . . 10 ⊢ (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))))
6	5	raleqbi1dv 3303	. . . . . . . . 9 ⊢ (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))))
7	4, 6	anbi12d 631	. . . . . . . 8 ⊢ (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)))))
8	3, 7	sbcie 3770	. . . . . . 7 ⊢ ([(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))))
9		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10		isghm.w	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑆)
11	9, 10	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
12	11	feq2d 6637	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶(Base‘𝑡)))
13		fveq2 6825	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
14		isghm.a	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑆)
15	13, 14	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
16	15	oveqd 7354	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑢(+g‘𝑠)𝑣) = (𝑢 + 𝑣))
17	16	fveqeq2d 6833	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))))
18	11, 17	raleqbidv 3315	. . . . . . . . 9 ⊢ (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))))
19	11, 18	raleqbidv 3315	. . . . . . . 8 ⊢ (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))))
20	12, 19	anbi12d 631	. . . . . . 7 ⊢ (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)))))
21	8, 20	bitrid 282	. . . . . 6 ⊢ (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)))))
22	21	abbidv 2805	. . . . 5 ⊢ (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)))})
23		fveq2 6825	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
24		isghm.x	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑇)
25	23, 24	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
26	25	feq3d 6638	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑓:𝑋⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶𝑌))
27		fveq2 6825	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
28		isghm.b	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝑇)
29	27, 28	eqtr4di 2794	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
30	29	oveqd 7354	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))
31	30	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))
32	31	2ralbidv 3208	. . . . . . 7 ⊢ (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))))
33	26, 32	anbi12d 631	. . . . . 6 ⊢ (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)))))
34	33	abbidv 2805	. . . . 5 ⊢ (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)))})
35	10	fvexi 6839	. . . . . . 7 ⊢ 𝑋 ∈ V
36	24	fvexi 6839	. . . . . . 7 ⊢ 𝑌 ∈ V
37		mapex 8692	. . . . . . 7 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V)
38	35, 36, 37	mp2an 689	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑋⟶𝑌} ∈ V
39		simpl 483	. . . . . . 7 ⊢ ((𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))) → 𝑓:𝑋⟶𝑌)
40	39	ss2abi 4011	. . . . . 6 ⊢ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ⊆ {𝑓 ∣ 𝑓:𝑋⟶𝑌}
41	38, 40	ssexi 5266	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))} ∈ V
42	22, 34, 1, 41	ovmpo 7495	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
43	42	eleq2d 2822	. . 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)))}))
44		fex 7158	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ∈ V) → 𝐹 ∈ V)
45	35, 44	mpan2 688	. . . . 5 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 ∈ V)
46	45	adantr 481	. . . 4 ⊢ ((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) → 𝐹 ∈ V)
47		feq1 6632	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋⟶𝑌 ↔ 𝐹:𝑋⟶𝑌))
48		fveq1 6824	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
49		fveq1 6824	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑢) = (𝐹‘𝑢))
50		fveq1 6824	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑣) = (𝐹‘𝑣))
51	49, 50	oveq12d 7355	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))
52	48, 51	eqeq12d 2752	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))
53	52	2ralbidv 3208	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))
54	47, 53	anbi12d 631	. . . 4 ⊢ (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)))))
55	46, 54	elab3 3627	. . 3 ⊢ (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))))
56	43, 55	bitrdi 286	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
57	2, 56	biadanii 819	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:   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2713  ∀wral 3061  Vcvv 3441  [wsbc 3727  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  +_gcplusg 17059  Grpcgrp 18673   GrpHom cghm 18927

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-ghm 18928

This theorem is referenced by:  isghm3  18931  ghmgrp1  18932  ghmgrp2  18933  ghmf  18934  ghmlin  18935  isghmd  18939  idghm  18945  ghmf1o  18960  rhmopp  20090  islmhm2  20406  expghm  20803  mulgghm2  20804  pi1xfr  24324  pi1coghm  24330  isrnghm  45809