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Mirrors > Home > MPE Home > Th. List > cayley | Structured version Visualization version GIF version |
Description: Cayley's Theorem (constructive version): given group 𝐺, 𝐹 is an isomorphism between 𝐺 and the subgroup 𝑆 of the symmetric group 𝐻 on the underlying set 𝑋 of 𝐺. See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
cayley.x | ⊢ 𝑋 = (Base‘𝐺) |
cayley.h | ⊢ 𝐻 = (SymGrp‘𝑋) |
cayley.p | ⊢ + = (+g‘𝐺) |
cayley.f | ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) |
cayley.s | ⊢ 𝑆 = ran 𝐹 |
Ref | Expression |
---|---|
cayley | ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐻) ∧ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)) ∧ 𝐹:𝑋–1-1-onto→𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cayley.s | . . 3 ⊢ 𝑆 = ran 𝐹 | |
2 | cayley.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
3 | cayley.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
4 | eqid 2622 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | cayley.h | . . . . 5 ⊢ 𝐻 = (SymGrp‘𝑋) | |
6 | eqid 2622 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | cayley.f | . . . . 5 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 2, 3, 4, 5, 6, 7 | cayleylem1 17832 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
9 | ghmrn 17673 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐺 ∈ Grp → ran 𝐹 ∈ (SubGrp‘𝐻)) |
11 | 1, 10 | syl5eqel 2705 | . 2 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐻)) |
12 | 1 | eqimss2i 3660 | . . . 4 ⊢ ran 𝐹 ⊆ 𝑆 |
13 | eqid 2622 | . . . . 5 ⊢ (𝐻 ↾s 𝑆) = (𝐻 ↾s 𝑆) | |
14 | 13 | resghm2b 17678 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐻) ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
15 | 11, 12, 14 | sylancl 694 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
16 | 8, 15 | mpbid 222 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆))) |
17 | 2, 3, 4, 5, 6, 7 | cayleylem2 17833 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→(Base‘𝐻)) |
18 | f1f1orn 6148 | . . . 4 ⊢ (𝐹:𝑋–1-1→(Base‘𝐻) → 𝐹:𝑋–1-1-onto→ran 𝐹) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1-onto→ran 𝐹) |
20 | f1oeq3 6129 | . . . 4 ⊢ (𝑆 = ran 𝐹 → (𝐹:𝑋–1-1-onto→𝑆 ↔ 𝐹:𝑋–1-1-onto→ran 𝐹)) | |
21 | 1, 20 | ax-mp 5 | . . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑆 ↔ 𝐹:𝑋–1-1-onto→ran 𝐹) |
22 | 19, 21 | sylibr 224 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1-onto→𝑆) |
23 | 11, 16, 22 | 3jca 1242 | 1 ⊢ (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐻) ∧ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)) ∧ 𝐹:𝑋–1-1-onto→𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ↦ cmpt 4729 ran crn 5115 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 0gc0g 16100 Grpcgrp 17422 SubGrpcsubg 17588 GrpHom cghm 17657 SymGrpcsymg 17797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-tset 15960 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-ghm 17658 df-ga 17723 df-symg 17798 |
This theorem is referenced by: cayleyth 17835 |
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