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Theorem kerf1ghm 19491
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 = (0g𝑅)
kerf1ghm.1 0 = (0g𝑆)
Assertion
Ref Expression
kerf1ghm (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))

Proof of Theorem kerf1ghm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 486 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵))
2 f1fn 6550 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
32adantl 485 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → 𝐹 Fn 𝐴)
4 elpreima 6805 . . . . . . . . . 10 (𝐹 Fn 𝐴 → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 })))
53, 4syl 17 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) ↔ (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 })))
65biimpa 480 . . . . . . . 8 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝑥𝐴 ∧ (𝐹𝑥) ∈ { 0 }))
76simpld 498 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → 𝑥𝐴)
86simprd 499 . . . . . . . 8 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹𝑥) ∈ { 0 })
9 fvex 6658 . . . . . . . . 9 (𝐹𝑥) ∈ V
109elsn 4540 . . . . . . . 8 ((𝐹𝑥) ∈ { 0 } ↔ (𝐹𝑥) = 0 )
118, 10sylib 221 . . . . . . 7 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → (𝐹𝑥) = 0 )
12 kerf1ghm.a . . . . . . . . . . 11 𝐴 = (Base‘𝑅)
13 kerf1ghm.b . . . . . . . . . . 11 𝐵 = (Base‘𝑆)
14 kerf1ghm.1 . . . . . . . . . . 11 0 = (0g𝑆)
15 kerf1ghm.n . . . . . . . . . . 11 𝑁 = (0g𝑅)
1612, 13, 14, 15f1ghm0to0 19488 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
1716biimpd 232 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
18173expa 1115 . . . . . . . 8 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 0𝑥 = 𝑁))
1918imp 410 . . . . . . 7 ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥𝐴) ∧ (𝐹𝑥) = 0 ) → 𝑥 = 𝑁)
201, 7, 11, 19syl21anc 836 . . . . . 6 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) ∧ 𝑥 ∈ (𝐹 “ { 0 })) → 𝑥 = 𝑁)
2120ex 416 . . . . 5 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) → 𝑥 = 𝑁))
22 velsn 4541 . . . . 5 (𝑥 ∈ {𝑁} ↔ 𝑥 = 𝑁)
2321, 22syl6ibr 255 . . . 4 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝑥 ∈ (𝐹 “ { 0 }) → 𝑥 ∈ {𝑁}))
2423ssrdv 3921 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝐹 “ { 0 }) ⊆ {𝑁})
25 ghmgrp1 18352 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp)
2612, 15grpidcl 18123 . . . . . . 7 (𝑅 ∈ Grp → 𝑁𝐴)
2725, 26syl 17 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁𝐴)
2815, 14ghmid 18356 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹𝑁) = 0 )
29 fvex 6658 . . . . . . . 8 (𝐹𝑁) ∈ V
3029elsn 4540 . . . . . . 7 ((𝐹𝑁) ∈ { 0 } ↔ (𝐹𝑁) = 0 )
3128, 30sylibr 237 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹𝑁) ∈ { 0 })
3212, 13ghmf 18354 . . . . . . 7 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:𝐴𝐵)
33 ffn 6487 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
34 elpreima 6805 . . . . . . 7 (𝐹 Fn 𝐴 → (𝑁 ∈ (𝐹 “ { 0 }) ↔ (𝑁𝐴 ∧ (𝐹𝑁) ∈ { 0 })))
3532, 33, 343syl 18 . . . . . 6 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝑁 ∈ (𝐹 “ { 0 }) ↔ (𝑁𝐴 ∧ (𝐹𝑁) ∈ { 0 })))
3627, 31, 35mpbir2and 712 . . . . 5 (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑁 ∈ (𝐹 “ { 0 }))
3736snssd 4702 . . . 4 (𝐹 ∈ (𝑅 GrpHom 𝑆) → {𝑁} ⊆ (𝐹 “ { 0 }))
3837adantr 484 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → {𝑁} ⊆ (𝐹 “ { 0 }))
3924, 38eqssd 3932 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵) → (𝐹 “ { 0 }) = {𝑁})
4032adantr 484 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴𝐵)
41 simpl 486 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
42 simpr2l 1229 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥𝐴)
43 simpr2r 1230 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑦𝐴)
44 simpr3 1193 . . . . . . . . . 10 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹𝑥) = (𝐹𝑦))
45 eqid 2798 . . . . . . . . . . . 12 (𝐹 “ { 0 }) = (𝐹 “ { 0 })
46 eqid 2798 . . . . . . . . . . . 12 (-g𝑅) = (-g𝑅)
4712, 14, 45, 46ghmeqker 18377 . . . . . . . . . . 11 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥𝐴𝑦𝐴) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 })))
4847biimpa 480 . . . . . . . . . 10 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦)) → (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 }))
4941, 42, 43, 44, 48syl31anc 1370 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) ∈ (𝐹 “ { 0 }))
50 simpr1 1191 . . . . . . . . 9 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝐹 “ { 0 }) = {𝑁})
5149, 50eleqtrd 2892 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) ∈ {𝑁})
52 ovex 7168 . . . . . . . . 9 (𝑥(-g𝑅)𝑦) ∈ V
5352elsn 4540 . . . . . . . 8 ((𝑥(-g𝑅)𝑦) ∈ {𝑁} ↔ (𝑥(-g𝑅)𝑦) = 𝑁)
5451, 53sylib 221 . . . . . . 7 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → (𝑥(-g𝑅)𝑦) = 𝑁)
5541, 25syl 17 . . . . . . . 8 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑅 ∈ Grp)
5612, 15, 46grpsubeq0 18177 . . . . . . . 8 ((𝑅 ∈ Grp ∧ 𝑥𝐴𝑦𝐴) → ((𝑥(-g𝑅)𝑦) = 𝑁𝑥 = 𝑦))
5755, 42, 43, 56syl3anc 1368 . . . . . . 7 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → ((𝑥(-g𝑅)𝑦) = 𝑁𝑥 = 𝑦))
5854, 57mpbid 235 . . . . . 6 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ((𝐹 “ { 0 }) = {𝑁} ∧ (𝑥𝐴𝑦𝐴) ∧ (𝐹𝑥) = (𝐹𝑦))) → 𝑥 = 𝑦)
59583anassrs 1357 . . . . 5 ((((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥𝐴𝑦𝐴)) ∧ (𝐹𝑥) = (𝐹𝑦)) → 𝑥 = 𝑦)
6059ex 416 . . . 4 (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) ∧ (𝑥𝐴𝑦𝐴)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6160ralrimivva 3156 . . 3 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
62 dff13 6991 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6340, 61, 62sylanbrc 586 . 2 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝐹 “ { 0 }) = {𝑁}) → 𝐹:𝐴1-1𝐵)
6439, 63impbida 800 1 (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wss 3881  {csn 4525  ccnv 5518  cima 5522   Fn wfn 6319  wf 6320  1-1wf1 6321  cfv 6324  (class class class)co 7135  Basecbs 16475  0gc0g 16705  Grpcgrp 18095  -gcsg 18097   GrpHom cghm 18347
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-ghm 18348
This theorem is referenced by:  dimkerim  31111  zrhf1ker  31326
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