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Theorem caovmo 6286
 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 6090 . . . . . 6
21eqeq1d 2446 . . . . 5
32mobidv 2318 . . . 4
4 oveq2 6091 . . . . . . 7
54eqeq1d 2446 . . . . . 6
65mo4 2316 . . . . 5
7 simpr 449 . . . . . . . . 9
87oveq2d 6099 . . . . . . . 8
9 simpl 445 . . . . . . . . . 10
109oveq1d 6098 . . . . . . . . 9
11 vex 2961 . . . . . . . . . . 11
12 vex 2961 . . . . . . . . . . 11
13 vex 2961 . . . . . . . . . . 11
14 caovmo.ass . . . . . . . . . . 11
1511, 12, 13, 14caovass 6249 . . . . . . . . . 10
16 caovmo.com . . . . . . . . . . 11
1711, 12, 13, 16, 14caov12 6277 . . . . . . . . . 10
1815, 17eqtri 2458 . . . . . . . . 9
19 caovmo.2 . . . . . . . . . . 11
2019elexi 2967 . . . . . . . . . 10
2120, 13, 16caovcom 6246 . . . . . . . . 9
2210, 18, 213eqtr3g 2493 . . . . . . . 8
238, 22eqtr3d 2472 . . . . . . 7
249, 19syl6eqel 2526 . . . . . . . . . 10
25 caovmo.dom . . . . . . . . . . 11
26 caovmo.3 . . . . . . . . . . 11
2725, 26ndmovrcl 6235 . . . . . . . . . 10
2824, 27syl 16 . . . . . . . . 9
2928simprd 451 . . . . . . . 8
30 oveq1 6090 . . . . . . . . . 10
31 id 21 . . . . . . . . . 10
3230, 31eqeq12d 2452 . . . . . . . . 9
33 caovmo.id . . . . . . . . 9
3432, 33vtoclga 3019 . . . . . . . 8
3529, 34syl 16 . . . . . . 7
367, 19syl6eqel 2526 . . . . . . . . . 10
3725, 26ndmovrcl 6235 . . . . . . . . . 10
3836, 37syl 16 . . . . . . . . 9
3938simprd 451 . . . . . . . 8
40 oveq1 6090 . . . . . . . . . 10
41 id 21 . . . . . . . . . 10
4240, 41eqeq12d 2452 . . . . . . . . 9
4342, 33vtoclga 3019 . . . . . . . 8
4439, 43syl 16 . . . . . . 7
4523, 35, 443eqtr3d 2478 . . . . . 6
4645ax-gen 1556 . . . . 5
476, 46mpgbir 1560 . . . 4
483, 47vtoclg 3013 . . 3
49 moanimv 2341 . . 3
5048, 49mpbir 202 . 2
51 eleq1 2498 . . . . . . 7
5219, 51mpbiri 226 . . . . . 6
5325, 26ndmovrcl 6235 . . . . . 6
5452, 53syl 16 . . . . 5
5554simpld 447 . . . 4
5655ancri 537 . . 3
5756moimi 2330 . 2
5850, 57ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360  wal 1550   wceq 1653   wcel 1726  wmo 2284  c0 3630   cxp 4878   cdm 4880  (class class class)co 6083 This theorem is referenced by:  recmulnq  8843 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-dm 4890  df-iota 5420  df-fv 5464  df-ov 6086
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