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Theorem caovmo 6512
 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:   ,,,   ,,,   ,,,   ,S,,   ,,,   ,,   ,   ,S

Proof of Theorem caovmo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6303 . . . . . 6
21eqeq1d 2459 . . . . 5
32mobidv 2305 . . . 4
4 oveq2 6304 . . . . . . 7
54eqeq1d 2459 . . . . . 6
65mo4 2337 . . . . 5
7 simpr 461 . . . . . . . . 9
87oveq2d 6312 . . . . . . . 8
9 simpl 457 . . . . . . . . . 10
109oveq1d 6311 . . . . . . . . 9
11 vex 3112 . . . . . . . . . . 11
12 vex 3112 . . . . . . . . . . 11
13 vex 3112 . . . . . . . . . . 11
14 caovmo.ass . . . . . . . . . . 11
1511, 12, 13, 14caovass 6475 . . . . . . . . . 10
16 caovmo.com . . . . . . . . . . 11
1711, 12, 13, 16, 14caov12 6503 . . . . . . . . . 10
1815, 17eqtri 2486 . . . . . . . . 9
19 caovmo.2 . . . . . . . . . . 11
2019elexi 3119 . . . . . . . . . 10
2120, 13, 16caovcom 6472 . . . . . . . . 9
2210, 18, 213eqtr3g 2521 . . . . . . . 8
238, 22eqtr3d 2500 . . . . . . 7
249, 19syl6eqel 2553 . . . . . . . . . 10
25 caovmo.dom . . . . . . . . . . 11
26 caovmo.3 . . . . . . . . . . 11
2725, 26ndmovrcl 6461 . . . . . . . . . 10
2824, 27syl 16 . . . . . . . . 9
2928simprd 463 . . . . . . . 8
30 oveq1 6303 . . . . . . . . . 10
31 id 22 . . . . . . . . . 10
3230, 31eqeq12d 2479 . . . . . . . . 9
33 caovmo.id . . . . . . . . 9
3432, 33vtoclga 3173 . . . . . . . 8
3529, 34syl 16 . . . . . . 7
367, 19syl6eqel 2553 . . . . . . . . . 10
3725, 26ndmovrcl 6461 . . . . . . . . . 10
3836, 37syl 16 . . . . . . . . 9
3938simprd 463 . . . . . . . 8
40 oveq1 6303 . . . . . . . . . 10
41 id 22 . . . . . . . . . 10
4240, 41eqeq12d 2479 . . . . . . . . 9
4342, 33vtoclga 3173 . . . . . . . 8
4439, 43syl 16 . . . . . . 7
4523, 35, 443eqtr3d 2506 . . . . . 6
4645ax-gen 1618 . . . . 5
476, 46mpgbir 1622 . . . 4
483, 47vtoclg 3167 . . 3
49 moanimv 2352 . . 3
5048, 49mpbir 209 . 2
51 eleq1 2529 . . . . . . 7
5219, 51mpbiri 233 . . . . . 6
5325, 26ndmovrcl 6461 . . . . . 6
5452, 53syl 16 . . . . 5
5554simpld 459 . . . 4
5655ancri 552 . . 3
5756moimi 2340 . 2
5850, 57ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:  -.wn 3  ->wi 4  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  E*wmo 2283   c0 3784  X.cxp 5002  domcdm 5004  (class class class)co 6296 This theorem is referenced by:  recmulnq  9363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-dm 5014  df-iota 5556  df-fv 5601  df-ov 6299
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