Theorem kerf1ghm 19317

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 487	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵))
2		f1fn 6776	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
3	2	adantl 486	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
4		elpreima 7054	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
5	3, 4	syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 })))
6	5	biimpa 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 }))
7	6	simpld 499	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 ∈ 𝐴)
8	6	simprd 500	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑥) ∈ { 0 })
9		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝑥) ∈ V
10	9	elsn 4609	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 )
11	8, 10	sylib 221	. . . . . . 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 19315	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
17	16	biimpd 232	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
18	17	3expa 1134	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
19	18	imp 411	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = 0 ) → 𝑥 = 𝑁)
20	1, 7, 11, 19	syl21anc 850	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 = 𝑁)
21	20	ex 417	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 = 𝑁))
22		velsn 4610	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
23	21, 22	imbitrrdi 255	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
24	23	ssrdv 3951	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) ⊆ {𝑁})
25		ghmgrp1 19288	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
26	12, 14	grpidcl 19032	. . . . . . 7 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
27	25, 26	syl 18	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
28	14, 15	ghmid 19292	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
29		fvex 6895	. . . . . . . 8 ⊢ (𝐹‘𝑁) ∈ V
30	29	elsn 4609	. . . . . . 7 ⊢ ((𝐹‘𝑁) ∈ { 0 } ↔ (𝐹‘𝑁) = 0 )
31	28, 30	sylibr 237	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) ∈ { 0 })
32	12, 13	ghmf 19290	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
33		ffn 6706	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
34		elpreima 7054	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
35	32, 33, 34	3syl 19	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑁 ∈ (◡𝐹 “ { 0 }) ↔ (𝑁 ∈ 𝐴 ∧ (𝐹‘𝑁) ∈ { 0 })))
36	27, 31, 35	mpbir2and 725	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ (◡𝐹 “ { 0 }))
37	36	snssd 4757	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
38	37	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
39	24, 38	eqssd 3962	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) = {𝑁})
40	32	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴⟶𝐵)
41		simpl 487	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
42		simpr2l 1249	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
43		simpr2r 1250	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
44		simpr3 1213	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
45		eqid 2769	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
46		eqid 2769	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
47	12, 15, 45, 46	ghmeqker 19313	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 })))
48	47	biimpa 481	. . . . . . . . . 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 1398	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
50		simpr1 1211	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (◡𝐹 “ { 0 }) = {𝑁})
51	49, 50	eleqtrd 2871	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ {𝑁})
52		ovex 7444	. . . . . . . . 9 ⊢ (𝑥(-g‘𝑅)𝑦) ∈ V
53	52	elsn 4609	. . . . . . . 8 ⊢ ((𝑥(-g‘𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g‘𝑅)𝑦) = 𝑁)
54	51, 53	sylib 221	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) = 𝑁)
55	25	adantr 485	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑅 ∈ Grp)
56	12, 14, 46	grpsubeq0 19092	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
57	55, 42, 43, 56	syl3anc 1396	. . . . . . 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 235	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	3anassrs 1379	. . . . 5 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
60	59	ex 417	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61	60	ralrimivva 3214	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
62		dff13 7253	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
63	40, 61, 62	sylanbrc 594	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴–1-1→𝐵)
64	39, 63	impbida 812	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  {csn 4594  ^◡ccnv 5661   “ cima 5665   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  0_gc0g 17492  Grpcgrp 19000  -_gcsg 19002   GrpHom cghm 19283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-sbg 19005  df-ghm 19284

This theorem is referenced by:  ghmqusker  19357  rngqiprngimf1  21411  dimkerim  33962  lvecendof1f1o  33968  zrhf1ker  34308