Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  caovmo Structured version   Unicode version

Theorem caovmo 6520
 Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)
Hypotheses
Ref Expression
caovmo.2
caovmo.dom
caovmo.3
caovmo.com
caovmo.ass
caovmo.id
Assertion
Ref Expression
caovmo
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,   ,   ,

Proof of Theorem caovmo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6312 . . . . . 6
21eqeq1d 2431 . . . . 5
32mobidv 2289 . . . 4
4 oveq2 6313 . . . . . . 7
54eqeq1d 2431 . . . . . 6
65mo4 2316 . . . . 5
7 simpr 462 . . . . . . . . 9
87oveq2d 6321 . . . . . . . 8
9 simpl 458 . . . . . . . . . 10
109oveq1d 6320 . . . . . . . . 9
11 vex 3090 . . . . . . . . . . 11
12 vex 3090 . . . . . . . . . . 11
13 vex 3090 . . . . . . . . . . 11
14 caovmo.ass . . . . . . . . . . 11
1511, 12, 13, 14caovass 6483 . . . . . . . . . 10
16 caovmo.com . . . . . . . . . . 11
1711, 12, 13, 16, 14caov12 6511 . . . . . . . . . 10
1815, 17eqtri 2458 . . . . . . . . 9
19 caovmo.2 . . . . . . . . . . 11
2019elexi 3097 . . . . . . . . . 10
2120, 13, 16caovcom 6480 . . . . . . . . 9
2210, 18, 213eqtr3g 2493 . . . . . . . 8
238, 22eqtr3d 2472 . . . . . . 7
249, 19syl6eqel 2525 . . . . . . . . . 10
25 caovmo.dom . . . . . . . . . . 11
26 caovmo.3 . . . . . . . . . . 11
2725, 26ndmovrcl 6469 . . . . . . . . . 10
2824, 27syl 17 . . . . . . . . 9
2928simprd 464 . . . . . . . 8
30 oveq1 6312 . . . . . . . . . 10
31 id 23 . . . . . . . . . 10
3230, 31eqeq12d 2451 . . . . . . . . 9
33 caovmo.id . . . . . . . . 9
3432, 33vtoclga 3151 . . . . . . . 8
3529, 34syl 17 . . . . . . 7
367, 19syl6eqel 2525 . . . . . . . . . 10
3725, 26ndmovrcl 6469 . . . . . . . . . 10
3836, 37syl 17 . . . . . . . . 9
3938simprd 464 . . . . . . . 8
40 oveq1 6312 . . . . . . . . . 10
41 id 23 . . . . . . . . . 10
4240, 41eqeq12d 2451 . . . . . . . . 9
4342, 33vtoclga 3151 . . . . . . . 8
4439, 43syl 17 . . . . . . 7
4523, 35, 443eqtr3d 2478 . . . . . 6
4645ax-gen 1665 . . . . 5
476, 46mpgbir 1669 . . . 4
483, 47vtoclg 3145 . . 3
49 moanimv 2330 . . 3
5048, 49mpbir 212 . 2
51 eleq1 2501 . . . . . . 7
5219, 51mpbiri 236 . . . . . 6
5325, 26ndmovrcl 6469 . . . . . 6
5452, 53syl 17 . . . . 5
5554simpld 460 . . . 4
5655ancri 554 . . 3
5756moimi 2319 . 2
5850, 57ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 370  wal 1435   wceq 1437   wcel 1870  wmo 2267  c0 3767   cxp 4852   cdm 4854  (class class class)co 6305 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-dm 4864  df-iota 5565  df-fv 5609  df-ov 6308 This theorem is referenced by:  recmulnq  9388
 Copyright terms: Public domain W3C validator