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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 2778 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | cayley.h | . . . . 5 ⊢ 𝐻 = (SymGrp‘𝑋) | |
6 | eqid 2778 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | cayley.f | . . . . 5 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 2, 3, 4, 5, 6, 7 | cayleylem1 18226 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
9 | ghmrn 18068 | . . . 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 2863 | . 2 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐻)) |
12 | 1 | eqimss2i 3879 | . . . 4 ⊢ ran 𝐹 ⊆ 𝑆 |
13 | eqid 2778 | . . . . 5 ⊢ (𝐻 ↾s 𝑆) = (𝐻 ↾s 𝑆) | |
14 | 13 | resghm2b 18073 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐻) ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
15 | 11, 12, 14 | sylancl 580 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
16 | 8, 15 | mpbid 224 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆))) |
17 | 2, 3, 4, 5, 6, 7 | cayleylem2 18227 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→(Base‘𝐻)) |
18 | f1f1orn 6404 | . . . 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 6384 | . . . 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 226 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1-onto→𝑆) |
23 | 11, 16, 22 | 3jca 1119 | 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 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ↦ cmpt 4967 ran crn 5358 –1-1→wf1 6134 –1-1-onto→wf1o 6136 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 ↾s cress 16267 +gcplusg 16349 0gc0g 16497 Grpcgrp 17820 SubGrpcsubg 17983 GrpHom cghm 18052 SymGrpcsymg 18191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-tset 16368 df-0g 16499 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-mhm 17732 df-submnd 17733 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-ghm 18053 df-ga 18117 df-symg 18192 |
This theorem is referenced by: cayleyth 18229 |
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