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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 2821 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | cayley.h | . . . . 5 ⊢ 𝐻 = (SymGrp‘𝑋) | |
6 | eqid 2821 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
7 | cayley.f | . . . . 5 ⊢ 𝐹 = (𝑔 ∈ 𝑋 ↦ (𝑎 ∈ 𝑋 ↦ (𝑔 + 𝑎))) | |
8 | 2, 3, 4, 5, 6, 7 | cayleylem1 18540 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
9 | ghmrn 18371 | . . . 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 2917 | . 2 ⊢ (𝐺 ∈ Grp → 𝑆 ∈ (SubGrp‘𝐻)) |
12 | 1 | eqimss2i 4026 | . . . 4 ⊢ ran 𝐹 ⊆ 𝑆 |
13 | eqid 2821 | . . . . 5 ⊢ (𝐻 ↾s 𝑆) = (𝐻 ↾s 𝑆) | |
14 | 13 | resghm2b 18376 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐻) ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
15 | 11, 12, 14 | sylancl 588 | . . 3 ⊢ (𝐺 ∈ Grp → (𝐹 ∈ (𝐺 GrpHom 𝐻) ↔ 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆)))) |
16 | 8, 15 | mpbid 234 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom (𝐻 ↾s 𝑆))) |
17 | 2, 3, 4, 5, 6, 7 | cayleylem2 18541 | . . . 4 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1→(Base‘𝐻)) |
18 | f1f1orn 6626 | . . . 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 6606 | . . . 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 236 | . 2 ⊢ (𝐺 ∈ Grp → 𝐹:𝑋–1-1-onto→𝑆) |
23 | 11, 16, 22 | 3jca 1124 | 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 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ↦ cmpt 5146 ran crn 5556 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 SubGrpcsubg 18273 GrpHom cghm 18355 SymGrpcsymg 18495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-submnd 17957 df-efmnd 18034 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-ga 18420 df-symg 18496 |
This theorem is referenced by: cayleyth 18543 |
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