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Theorem caovimo 5636
Description: Uniqueness of inverse element in commutative, associative operation with identity. The identity element is . (Contributed by Jim Kingdon, 18-Sep-2019.)
Hypotheses
Ref Expression
caovimo.idel  S
caovimo.com  S  S  F  F
caovimo.ass  S  S  S  F F  F F
caovimo.id  S  F
Assertion
Ref Expression
caovimo  S  S  F
Distinct variable groups:   ,,,,   ,,,   , F,,,   , S,,,
Allowed substitution hint:   ()

Proof of Theorem caovimo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq1 5462 . . . . . . 7  F  F F  F
21adantl 262 . . . . . 6  S  F  F F  F
323ad2ant2 925 . . . . 5  S  S  F  S  F  F F  F
4 df-3an 886 . . . . . . . . 9  S  S  S  S  S  S
5 caovimo.ass . . . . . . . . . . . . . 14  S  S  S  F F  F F
65adantl 262 . . . . . . . . . . . . 13  S  S  S  S  S  S  F F  F F
7 simp1 903 . . . . . . . . . . . . 13  S  S  S  S
8 simp2 904 . . . . . . . . . . . . 13  S  S  S  S
9 simp3 905 . . . . . . . . . . . . 13  S  S  S  S
106, 7, 8, 9caovassd 5602 . . . . . . . . . . . 12  S  S  S  F F  F F
11 caovimo.com . . . . . . . . . . . . . 14  S  S  F  F
1211adantl 262 . . . . . . . . . . . . 13  S  S  S  S  S  F  F
137, 8, 9, 12, 6caov12d 5624 . . . . . . . . . . . 12  S  S  S  F F  F F
1410, 13eqtrd 2069 . . . . . . . . . . 11  S  S  S  F F  F F
1514adantr 261 . . . . . . . . . 10  S  S  S  F  F F  F F
16 oveq2 5463 . . . . . . . . . . . 12  F  F F  F
17 oveq1 5462 . . . . . . . . . . . . . 14  F  F
18 id 19 . . . . . . . . . . . . . 14
1917, 18eqeq12d 2051 . . . . . . . . . . . . 13  F  F
20 caovimo.id . . . . . . . . . . . . 13  S  F
2119, 20vtoclga 2613 . . . . . . . . . . . 12  S  F
2216, 21sylan9eqr 2091 . . . . . . . . . . 11  S  F  F F
23223ad2antl2 1066 . . . . . . . . . 10  S  S  S  F  F F
2415, 23eqtrd 2069 . . . . . . . . 9  S  S  S  F  F F
254, 24sylanbr 269 . . . . . . . 8  S  S  S  F  F F
2625anasss 379 . . . . . . 7  S  S  S  F  F F
27263impa 1098 . . . . . 6  S  S  S  F  F F
28273adant2r 1129 . . . . 5  S  S  F  S  F  F F
2911adantl 262 . . . . . . . . 9  S  S  S  F  F
30 caovimo.idel . . . . . . . . . 10  S
3130a1i 9 . . . . . . . . 9  S  S
32 id 19 . . . . . . . . 9  S  S
3329, 31, 32caovcomd 5599 . . . . . . . 8  S  F  F
34 oveq1 5462 . . . . . . . . . 10  F  F
35 id 19 . . . . . . . . . 10
3634, 35eqeq12d 2051 . . . . . . . . 9  F  F
3736, 20vtoclga 2613 . . . . . . . 8  S  F
3833, 37eqtrd 2069 . . . . . . 7  S  F
3938adantr 261 . . . . . 6  S  F  F
40393ad2ant3 926 . . . . 5  S  S  F  S  F  F
413, 28, 403eqtr3d 2077 . . . 4  S  S  F  S  F
42413expib 1106 . . 3  S  S  F  S  F
4342alrimivv 1752 . 2  S  S  F  S  F
44 eleq1 2097 . . . 4  S  S
45 oveq2 5463 . . . . 5  F  F
4645eqeq1d 2045 . . . 4  F  F
4744, 46anbi12d 442 . . 3  S  F  S  F
4847mo4 1958 . 2  S  F  S  F  S  F
4943, 48sylibr 137 1  S  S  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884  wal 1240   wceq 1242   wcel 1390  wmo 1898  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  recmulnqg  6375
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