Theorem sn-isghm 43463

Description: Longer proof of isghm 19330, unsuccessfully attempting to simplify isghm 19330 using elovmpo 7665 according to an editorial note (now removed). (Contributed by SN, 7-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
sn-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 sn-isghm

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

Step	Hyp	Ref	Expression
1		df-ghm 19328	. . 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		fvex 6898	. . . . . 6 ⊢ (Base‘𝑠) ∈ V
3		feq2 6688	. . . . . . 7 ⊢ (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
4		raleq 3322	. . . . . . . 8 ⊢ (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))))
5	4	raleqbi1dv 3335	. . . . . . 7 ⊢ (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))))
6	3, 5	anbi12d 644	. . . . . 6 ⊢ (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)))))
7	2, 6	sbcie 3787	. . . . 5 ⊢ ([(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))))
8	7	abbii 2832	. . . 4 ⊢ {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:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))}
9		fvex 6898	. . . . . 6 ⊢ (Base‘𝑡) ∈ V
10		fsetex 8859	. . . . . 6 ⊢ ((Base‘𝑡) ∈ V → {𝑓 ∣ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)} ∈ V)
11	9, 10	ax-mp 5	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)} ∈ V
12		abanssl 4264	. . . . 5 ⊢ {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} ⊆ {𝑓 ∣ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)}
13	11, 12	ssexi 5295	. . . 4 ⊢ {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} ∈ V
14	8, 13	eqeltri 2861	. . 3 ⊢ {𝑓 ∣ [(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢 ∈ 𝑤 ∀𝑣 ∈ 𝑤 (𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} ∈ V
15		fveq2 6885	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
16		sn-isghm.w	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑆)
17	15, 16	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
18	17	adantr 486	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝑋)
19		fveq2 6885	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
20		sn-isghm.x	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑇)
21	19, 20	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
22	21	adantl 487	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑡) = 𝑌)
23	18, 22	feq23d 6704	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶𝑌))
24		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
25		sn-isghm.a	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑆)
26	24, 25	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
27	26	oveqd 7436	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑢(+g‘𝑠)𝑣) = (𝑢 + 𝑣))
28	27	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑢(+g‘𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣)))
29		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
30		sn-isghm.b	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝑇)
31	29, 30	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
32	31	oveqd 7436	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))
33	28, 32	eqeqan12d 2779	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))
34	18, 33	raleqbidv 3340	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)) ↔ ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣))))
35	18, 34	raleqbidv 3340	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀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) ⨣ (𝑓‘𝑣))))
36	23, 35	anbi12d 644	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣))) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))))
37	36	abbidv 2831	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g‘𝑠)𝑣)) = ((𝑓‘𝑢)(+g‘𝑡)(𝑓‘𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
38	8, 37	eqtrid 2812	. . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {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⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))})
39	1, 14, 38	elovmpo 7665	. 2 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))}))
40	16	fvexi 6899	. . . . . 6 ⊢ 𝑋 ∈ V
41	20	fvexi 6899	. . . . . 6 ⊢ 𝑌 ∈ V
42		fex2 7939	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐹 ∈ V)
43	40, 41, 42	mp3an23 1482	. . . . 5 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 ∈ V)
44	43	adantr 486	. . . 4 ⊢ ((𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))) → 𝐹 ∈ V)
45		feq1 6687	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋⟶𝑌 ↔ 𝐹:𝑋⟶𝑌))
46		fveq1 6884	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
47		fveq1 6884	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑢) = (𝐹‘𝑢))
48		fveq1 6884	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑣) = (𝐹‘𝑣))
49	47, 48	oveq12d 7437	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))
50	46, 49	eqeq12d 2781	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣))))
51	50	2ralbidv 3231	. . . . 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))))
52	45, 51	anbi12d 644	. . . 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)))))
53	44, 52	elab3 3647	. . 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))))
54	53	3anbi3i 1177	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓‘𝑢) ⨣ (𝑓‘𝑣)))}) ↔ (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
55		df-3an 1105	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
56	39, 54, 55	3bitri 300	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  Vcvv 3457  [wsbc 3746  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  Grpcgrp 19044   GrpHom cghm 19327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-ghm 19328

This theorem is used by: (None)