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Theorem ecopovtrng 6206
 Description: Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by Jim Kingdon, 1-Sep-2019.)
Hypotheses
Ref Expression
ecopopr.1
ecopoprg.com
ecopoprg.cl
ecopoprg.ass
ecopoprg.can
Assertion
Ref Expression
ecopovtrng
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)   (,,,,,)   (,,,,,)   (,,,,,)

Proof of Theorem ecopovtrng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7
2 opabssxp 4414 . . . . . . 7
31, 2eqsstri 2975 . . . . . 6
43brel 4392 . . . . 5
54simpld 105 . . . 4
63brel 4392 . . . 4
75, 6anim12i 321 . . 3
8 3anass 889 . . 3
97, 8sylibr 137 . 2
10 eqid 2040 . . 3
11 breq1 3767 . . . . 5
1211anbi1d 438 . . . 4
13 breq1 3767 . . . 4
1412, 13imbi12d 223 . . 3
15 breq2 3768 . . . . 5
16 breq1 3767 . . . . 5
1715, 16anbi12d 442 . . . 4
1817imbi1d 220 . . 3
19 breq2 3768 . . . . 5
2019anbi2d 437 . . . 4
21 breq2 3768 . . . 4
2220, 21imbi12d 223 . . 3
231ecopoveq 6201 . . . . . . . 8
24233adant3 924 . . . . . . 7
251ecopoveq 6201 . . . . . . . 8
26253adant1 922 . . . . . . 7
2724, 26anbi12d 442 . . . . . 6
28 oveq12 5521 . . . . . . 7
29 simp2l 930 . . . . . . . . 9
30 simp2r 931 . . . . . . . . 9
31 simp1l 928 . . . . . . . . 9
32 ecopoprg.com . . . . . . . . . 10
3332adantl 262 . . . . . . . . 9
34 ecopoprg.ass . . . . . . . . . 10
3534adantl 262 . . . . . . . . 9
36 simp3r 933 . . . . . . . . 9
37 ecopoprg.cl . . . . . . . . . 10
3837adantl 262 . . . . . . . . 9
3929, 30, 31, 33, 35, 36, 38caov411d 5686 . . . . . . . 8
40 simp1r 929 . . . . . . . . . 10
41 simp3l 932 . . . . . . . . . 10
4240, 30, 29, 33, 35, 41, 38caov411d 5686 . . . . . . . . 9
4340, 30, 29, 33, 35, 41, 38caov4d 5685 . . . . . . . . 9
4442, 43eqtr3d 2074 . . . . . . . 8
4539, 44eqeq12d 2054 . . . . . . 7
4628, 45syl5ibr 145 . . . . . 6
4727, 46sylbid 139 . . . . 5
48 ecopoprg.can . . . . . . . 8
49 oveq2 5520 . . . . . . . 8
5048, 49impbid1 130 . . . . . . 7
5150adantl 262 . . . . . 6
5237caovcl 5655 . . . . . . 7
5329, 30, 52syl2anc 391 . . . . . 6
5437caovcl 5655 . . . . . . 7
5531, 36, 54syl2anc 391 . . . . . 6
5638, 40, 41caovcld 5654 . . . . . 6
5751, 53, 55, 56caovcand 5663 . . . . 5
5847, 57sylibd 138 . . . 4
591ecopoveq 6201 . . . . 5
60593adant2 923 . . . 4
6158, 60sylibrd 158 . . 3
6210, 14, 18, 22, 613optocl 4418 . 2
639, 62mpcom 32 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   w3a 885   wceq 1243  wex 1381   wcel 1393  cop 3378   class class class wbr 3764  copab 3817   cxp 4343  (class class class)co 5512 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515 This theorem is referenced by:  ecopoverg  6207
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