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Theorem ecopovtrn 6269
 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 NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
ecopopr.1
ecopopr.com
ecopopr.cl
ecopopr.ass
ecopopr.can
Assertion
Ref Expression
ecopovtrn
Distinct variable groups:   ,,,,,,   ,,,,,,
Allowed substitution hints:   (,,,,,)   (,,,,,)   (,,,,,)   (,,,,,)

Proof of Theorem ecopovtrn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7
2 opabssxp 4440 . . . . . . 7
31, 2eqsstri 3030 . . . . . 6
43brel 4418 . . . . 5
54simpld 110 . . . 4
63brel 4418 . . . 4
75, 6anim12i 331 . . 3
8 3anass 924 . . 3
97, 8sylibr 132 . 2
10 eqid 2082 . . 3
11 breq1 3796 . . . . 5
1211anbi1d 453 . . . 4
13 breq1 3796 . . . 4
1412, 13imbi12d 232 . . 3
15 breq2 3797 . . . . 5
16 breq1 3796 . . . . 5
1715, 16anbi12d 457 . . . 4
1817imbi1d 229 . . 3
19 breq2 3797 . . . . 5
2019anbi2d 452 . . . 4
21 breq2 3797 . . . 4
2220, 21imbi12d 232 . . 3
231ecopoveq 6267 . . . . . . . 8
24233adant3 959 . . . . . . 7
251ecopoveq 6267 . . . . . . . 8
26253adant1 957 . . . . . . 7
2724, 26anbi12d 457 . . . . . 6
28 oveq12 5552 . . . . . . 7
29 simp2l 965 . . . . . . . . 9
30 simp2r 966 . . . . . . . . 9
31 simp1l 963 . . . . . . . . 9
32 ecopopr.com . . . . . . . . . 10
3332a1i 9 . . . . . . . . 9
34 ecopopr.ass . . . . . . . . . 10
3534a1i 9 . . . . . . . . 9
36 simp3r 968 . . . . . . . . 9
37 ecopopr.cl . . . . . . . . . 10
3837adantl 271 . . . . . . . . 9
3929, 30, 31, 33, 35, 36, 38caov411d 5717 . . . . . . . 8
40 simp1r 964 . . . . . . . . . 10
41 simp3l 967 . . . . . . . . . 10
4240, 30, 29, 33, 35, 41, 38caov411d 5717 . . . . . . . . 9
4340, 30, 29, 33, 35, 41, 38caov4d 5716 . . . . . . . . 9
4442, 43eqtr3d 2116 . . . . . . . 8
4539, 44eqeq12d 2096 . . . . . . 7
4628, 45syl5ibr 154 . . . . . 6
4727, 46sylbid 148 . . . . 5
48 ecopopr.can . . . . . . . . 9
49483adant3 959 . . . . . . . 8
50 oveq2 5551 . . . . . . . 8
5149, 50impbid1 140 . . . . . . 7
5251adantl 271 . . . . . 6
5337caovcl 5686 . . . . . . 7
5429, 30, 53syl2anc 403 . . . . . 6
5537caovcl 5686 . . . . . . 7
5631, 36, 55syl2anc 403 . . . . . 6
5738, 40, 41caovcld 5685 . . . . . 6
5852, 54, 56, 57caovcand 5694 . . . . 5
5947, 58sylibd 147 . . . 4
601ecopoveq 6267 . . . . 5
61603adant2 958 . . . 4
6259, 61sylibrd 167 . . 3
6310, 14, 18, 22, 623optocl 4444 . 2
649, 63mpcom 36 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   w3a 920   wceq 1285  wex 1422   wcel 1434  cop 3409   class class class wbr 3793  copab 3846   cxp 4369  (class class class)co 5543 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-iota 4897  df-fv 4940  df-ov 5546 This theorem is referenced by:  ecopover  6270
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