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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 2740 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | cayley.h | . . . . 5 ⊢ 𝐻 = (SymGrp‘𝑋) | |
6 | eqid 2740 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | cayley.f | . . . . 5 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 2, 3, 4, 5, 6, 7 | cayleylem1 19018 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
9 | ghmrn 18845 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐺 ∈ Grp → ran 𝐹 ∈ (SubGrp‘𝐻)) |
11 | 1, 10 | eqeltrid 2845 | . 2 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐻)) |
12 | 1 | eqimss2i 3985 | . . . 4 ⊢ ran 𝐹 ⊆ 𝑆 |
13 | eqid 2740 | . . . . 5 ⊢ (𝐻 ↾s 𝑆) = (𝐻 ↾s 𝑆) | |
14 | 13 | resghm2b 18850 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐻) ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
15 | 11, 12, 14 | sylancl 586 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
16 | 8, 15 | mpbid 231 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆))) |
17 | 2, 3, 4, 5, 6, 7 | cayleylem2 19019 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→(Base‘𝐻)) |
18 | f1f1orn 6725 | . . . 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 6704 | . . . 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 233 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1-onto→𝑆) |
23 | 11, 16, 22 | 3jca 1127 | 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 205 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ↦ cmpt 5162 ran crn 5591 –1-1→wf1 6429 –1-1-onto→wf1o 6431 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 ↾s cress 16939 +gcplusg 16960 0gc0g 17148 Grpcgrp 18575 SubGrpcsubg 18747 GrpHom cghm 18829 SymGrpcsymg 18972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-tset 16979 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-mhm 18428 df-submnd 18429 df-efmnd 18506 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-ghm 18830 df-ga 18894 df-symg 18973 |
This theorem is referenced by: cayleyth 19021 |
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