Theorem kerf1ghm 19191

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 482	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵))
2		f1fn 6739	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
3	2	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
4		elpreima 7012	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
6	5	biimpa 476	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 }))
7	6	simpld 494	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 ∈ 𝐴)
8	6	simprd 495	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑥) ∈ { 0 })
9		fvex 6855	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
10	9	elsn 4597	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 )
11	8, 10	sylib 218	. . . . . . 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 19189	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
17	16	biimpd 229	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
18	17	3expa 1119	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
19	18	imp 406	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = 0 ) → 𝑥 = 𝑁)
20	1, 7, 11, 19	syl21anc 838	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 = 𝑁)
21	20	ex 412	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 = 𝑁))
22		velsn 4598	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
23	21, 22	imbitrrdi 252	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
24	23	ssrdv 3941	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) ⊆ {𝑁})
25		ghmgrp1 19162	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
26	12, 14	grpidcl 18910	. . . . . . 7 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
28	14, 15	ghmid 19166	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
29		fvex 6855	. . . . . . . 8 ⊢ (𝐹‘𝑁) ∈ V
30	29	elsn 4597	. . . . . . 7 ⊢ ((𝐹‘𝑁) ∈ { 0 } ↔ (𝐹‘𝑁) = 0 )
31	28, 30	sylibr 234	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) ∈ { 0 })
32	12, 13	ghmf 19164	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
33		ffn 6670	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
34		elpreima 7012	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
35	32, 33, 34	3syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
36	27, 31, 35	mpbir2and 714	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ (◡𝐹 “ { 0 }))
37	36	snssd 4767	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
38	37	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
39	24, 38	eqssd 3953	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) = {𝑁})
40	32	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴⟶𝐵)
41		simpl 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
42		simpr2l 1234	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
43		simpr2r 1235	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
44		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
45		eqid 2737	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
46		eqid 2737	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
47	12, 15, 45, 46	ghmeqker 19187	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 })))
48	47	biimpa 476	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
49	41, 42, 43, 44, 48	syl31anc 1376	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
50		simpr1 1196	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (◡𝐹 “ { 0 }) = {𝑁})
51	49, 50	eleqtrd 2839	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ {𝑁})
52		ovex 7401	. . . . . . . . 9 ⊢ (𝑥(-g‘𝑅)𝑦) ∈ V
53	52	elsn 4597	. . . . . . . 8 ⊢ ((𝑥(-g‘𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g‘𝑅)𝑦) = 𝑁)
54	51, 53	sylib 218	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) = 𝑁)
55	25	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑅 ∈ Grp)
56	12, 14, 46	grpsubeq0 18971	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
57	55, 42, 43, 56	syl3anc 1374	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
58	54, 57	mpbid 232	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	3anassrs 1362	. . . . 5 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
60	59	ex 412	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61	60	ralrimivva 3181	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
62		dff13 7210	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
63	40, 61, 62	sylanbrc 584	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴–1-1→𝐵)
64	39, 63	impbida 801	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  {csn 4582  ^◡ccnv 5631   “ cima 5635   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  0_gc0g 17371  Grpcgrp 18878  -_gcsg 18880   GrpHom cghm 19156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18881  df-minusg 18882  df-sbg 18883  df-ghm 19157

This theorem is referenced by:  ghmqusker  19231  rngqiprngimf1  21270  dimkerim  33809  lvecendof1f1o  33815  zrhf1ker  34155