Theorem kerf1ghm 13347

Description: A group homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)

Hypotheses

Ref	Expression
f1ghm0to0.a	⊢ 𝐴 = (Base‘𝑅)
f1ghm0to0.b	⊢ 𝐵 = (Base‘𝑆)
f1ghm0to0.n	⊢ 𝑁 = (0g‘𝑅)
f1ghm0to0.0	⊢ 0 = (0g‘𝑆)

Assertion

Ref	Expression
kerf1ghm	⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))

Proof of Theorem kerf1ghm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵))
2		f1fn 5462	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
3	2	adantl 277	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
4		elpreima 5678	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
5	3, 4	syl 14	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
6	5	biimpa 296	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 }))
7	6	simpld 112	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 ∈ 𝐴)
8	6	simprd 114	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑥) ∈ { 0 })
9		elsng 3634	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ { 0 } → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
10	8, 9	syl 14	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 ))
11	8, 10	mpbid 147	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑥) = 0 )
12		f1ghm0to0.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Base‘𝑅)
13		f1ghm0to0.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑆)
14		f1ghm0to0.n	. . . . . . . . . . 11 ⊢ 𝑁 = (0g‘𝑅)
15		f1ghm0to0.0	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
16	12, 13, 14, 15	f1ghm0to0 13345	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
17	16	biimpd 144	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
18	17	3expa 1205	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
19	18	imp 124	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = 0 ) → 𝑥 = 𝑁)
20	1, 7, 11, 19	syl21anc 1248	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 = 𝑁)
21	20	ex 115	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 = 𝑁))
22		velsn 3636	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
23	21, 22	imbitrrdi 162	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
24	23	ssrdv 3186	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) ⊆ {𝑁})
25		ghmgrp1 13318	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
26	12, 14	grpidcl 13104	. . . . . . 7 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
27	25, 26	syl 14	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
28	14, 15	ghmid 13322	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
29	12, 13	ghmf 13320	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
30	29, 27	ffvelcdmd 5695	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) ∈ 𝐵)
31		elsng 3634	. . . . . . . 8 ⊢ ((𝐹‘𝑁) ∈ 𝐵 → ((𝐹‘𝑁) ∈ { 0 } ↔ (𝐹‘𝑁) = 0 ))
32	30, 31	syl 14	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → ((𝐹‘𝑁) ∈ { 0 } ↔ (𝐹‘𝑁) = 0 ))
33	28, 32	mpbird 167	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) ∈ { 0 })
34		ffn 5404	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
35		elpreima 5678	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
36	29, 34, 35	3syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
37	27, 33, 36	mpbir2and 946	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ (◡𝐹 “ { 0 }))
38	37	snssd 3764	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
39	38	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
40	24, 39	eqssd 3197	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) = {𝑁})
41	29	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴⟶𝐵)
42		simpl 109	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
43		simpr2l 1058	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
44		simpr2r 1059	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
45		simpr3 1007	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
46		eqid 2193	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
47		eqid 2193	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
48	12, 15, 46, 47	ghmeqker 13344	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 })))
49	48	biimpa 296	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
50	42, 43, 44, 45, 49	syl31anc 1252	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
51		simpr1 1005	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (◡𝐹 “ { 0 }) = {𝑁})
52	50, 51	eleqtrd 2272	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ {𝑁})
53		simp2 1000	. . . . . . . . . 10 ⊢ (((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
54	12, 47	grpsubcl 13155	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥(-g‘𝑅)𝑦) ∈ 𝐴)
55	54	3expb 1206	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥(-g‘𝑅)𝑦) ∈ 𝐴)
56	25, 53, 55	syl2an 289	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ 𝐴)
57		elsng 3634	. . . . . . . . 9 ⊢ ((𝑥(-g‘𝑅)𝑦) ∈ 𝐴 → ((𝑥(-g‘𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g‘𝑅)𝑦) = 𝑁))
58	56, 57	syl 14	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥(-g‘𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g‘𝑅)𝑦) = 𝑁))
59	52, 58	mpbid 147	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) = 𝑁)
60	25	adantr 276	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑅 ∈ Grp)
61	12, 14, 47	grpsubeq0 13161	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
62	60, 43, 44, 61	syl3anc 1249	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
63	59, 62	mpbid 147	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
64	63	3anassrs 1231	. . . . 5 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
65	64	ex 115	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
66	65	ralrimivva 2576	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
67		dff13 5812	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
68	41, 66, 67	sylanbrc 417	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴–1-1→𝐵)
69	40, 68	impbida 596	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3154  {csn 3619  ^◡ccnv 4659   “ cima 4663   Fn wfn 5250  ⟶wf 5251  –1-1→wf1 5252  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  0_gc0g 12870  Grpcgrp 13075  -_gcsg 13077   GrpHom cghm 13313

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-sbg 13080  df-ghm 13314

This theorem is referenced by: (None)