Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  caytr Unicode version

Theorem caytr 25503
 Description: "It follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor, the effect is simply to reproduce the group... ." Cayley, On the theory of groups, as depending on the symbolic equation th^n = 1, 1854. (it is the original paper where the axiomatic definition of a group was given for the first time.) (Contributed by FL, 15-Oct-2012.)
Hypotheses
Ref Expression
trfun.2
trinv.1
caytr.1
Assertion
Ref Expression
caytr
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem caytr
StepHypRef Expression
1 ssid 3210 . . . 4
2 trinv.1 . . . . . . 7
32unsgrp 25470 . . . . . 6
43adantr 451 . . . . 5
5 elpwg 3645 . . . . 5
64, 5syl 15 . . . 4
71, 6mpbiri 224 . . 3
8 trfun.2 . . . 4
9 caytr.1 . . . 4
108, 2, 9prsubrtr 25502 . . 3
117, 10mpd3an3 1278 . 2
128, 2trooo 25497 . . 3
13 f1ofn 5489 . . 3
14 fnima 5378 . . 3
1512, 13, 143syl 18 . 2
168, 2trran2 25496 . 2
1711, 15, 163eqtrd 2332 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1632   wcel 1696  cvv 2801   wss 3165  cpw 3638  csn 3653   cmpt 4093   crn 4706  cima 4708   wfn 5266  wf1o 5270  cfv 5271  (class class class)co 5874  cgr 20869  ccst 25275 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-cst 25276
 Copyright terms: Public domain W3C validator