Theorem kerf1ghm 20177

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
kerf1ghm.a	⊢ 𝐴 = (Base‘𝑅)
kerf1ghm.b	⊢ 𝐵 = (Base‘𝑆)
kerf1ghm.n	⊢ 𝑁 = (0g‘𝑅)
kerf1ghm.1	⊢ 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 483	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 Fn 𝐴)
4		elpreima 7008	. . . . . . . . . 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 477	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) ∈ { 0 }))
7	6	simpld 495	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 ∈ 𝐴)
8	6	simprd 496	. . . . . . . 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 4601	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ { 0 } ↔ (𝐹‘𝑥) = 0 )
11	8, 10	sylib 217	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → (𝐹‘𝑥) = 0 )
12		kerf1ghm.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Base‘𝑅)
13		kerf1ghm.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑆)
14		kerf1ghm.1	. . . . . . . . . . 11 ⊢ 0 = (0g‘𝑆)
15		kerf1ghm.n	. . . . . . . . . . 11 ⊢ 𝑁 = (0g‘𝑅)
16	12, 13, 14, 15	f1ghm0to0 20174	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 𝑁))
17	16	biimpd 228	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
18	17	3expa 1118	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 0 → 𝑥 = 𝑁))
19	18	imp 407	. . . . . . 7 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ 𝐴) ∧ (𝐹‘𝑥) = 0 ) → 𝑥 = 𝑁)
20	1, 7, 11, 19	syl21anc 836	. . . . . 6 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑥 ∈ (◡𝐹 “ { 0 })) → 𝑥 = 𝑁)
21	20	ex 413	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 = 𝑁))
22		velsn 4602	. . . . 5 ⊢ (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
23	21, 22	syl6ibr 251	. . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (𝑥 ∈ (◡𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
24	23	ssrdv 3950	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) ⊆ {𝑁})
25		ghmgrp1 19010	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
26	12, 15	grpidcl 18778	. . . . . . 7 ⊢ (𝑅 ∈ Grp → 𝑁 ∈ 𝐴)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ 𝐴)
28	15, 14	ghmid 19014	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) = 0 )
29		fvex 6855	. . . . . . . 8 ⊢ (𝐹‘𝑁) ∈ V
30	29	elsn 4601	. . . . . . 7 ⊢ ((𝐹‘𝑁) ∈ { 0 } ↔ (𝐹‘𝑁) = 0 )
31	28, 30	sylibr 233	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘𝑁) ∈ { 0 })
32	12, 13	ghmf 19012	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴⟶𝐵)
33		ffn 6668	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
34		elpreima 7008	. . . . . . 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 711	. . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ (◡𝐹 “ { 0 }))
37	36	snssd 4769	. . . 4 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
38	37	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → {𝑁} ⊆ (◡𝐹 “ { 0 }))
39	24, 38	eqssd 3961	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) → (◡𝐹 “ { 0 }) = {𝑁})
40	32	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴⟶𝐵)
41		simpl 483	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
42		simpr2l 1232	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
43		simpr2r 1233	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑦 ∈ 𝐴)
44		simpr3 1196	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝐹‘𝑥) = (𝐹‘𝑦))
45		eqid 2736	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ { 0 }) = (◡𝐹 “ { 0 })
46		eqid 2736	. . . . . . . . . . . 12 ⊢ (-g‘𝑅) = (-g‘𝑅)
47	12, 14, 45, 46	ghmeqker 19035	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 })))
48	47	biimpa 477	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ (◡𝐹 “ { 0 }))
50		simpr1 1194	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (◡𝐹 “ { 0 }) = {𝑁})
51	49, 50	eleqtrd 2840	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) ∈ {𝑁})
52		ovex 7390	. . . . . . . . 9 ⊢ (𝑥(-g‘𝑅)𝑦) ∈ V
53	52	elsn 4601	. . . . . . . 8 ⊢ ((𝑥(-g‘𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g‘𝑅)𝑦) = 𝑁)
54	51, 53	sylib 217	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → (𝑥(-g‘𝑅)𝑦) = 𝑁)
55	41, 25	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑅 ∈ Grp)
56	12, 15, 46	grpsubeq0 18833	. . . . . . . 8 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥(-g‘𝑅)𝑦) = 𝑁 ↔ 𝑥 = 𝑦))
57	55, 42, 43, 56	syl3anc 1371	. . . . . . 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 231	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((◡𝐹 “ { 0 }) = {𝑁} ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐹‘𝑥) = (𝐹‘𝑦))) → 𝑥 = 𝑦)
59	58	3anassrs 1360	. . . . 5 ⊢ ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝐹‘𝑥) = (𝐹‘𝑦)) → 𝑥 = 𝑦)
60	59	ex 413	. . . 4 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
61	60	ralrimivva 3197	. . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
62		dff13 7202	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
63	40, 61, 62	sylanbrc 583	. 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (◡𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴–1-1→𝐵)
64	39, 63	impbida 799	1 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3910  {csn 4586  ^◡ccnv 5632   “ cima 5636   Fn wfn 6491  ⟶wf 6492  –1-1→wf1 6493  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  0_gc0g 17321  Grpcgrp 18748  -_gcsg 18750   GrpHom cghm 19005

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-ghm 19006

This theorem is referenced by:  ghmqusker  32198  dimkerim  32322  zrhf1ker  32556