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Theorem ecopovtrn 6966
 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 4909 . . . . . . 7
31, 2eqsstri 3338 . . . . . 6
43brel 4885 . . . . 5
54simpld 446 . . . 4
63brel 4885 . . . 4
75, 6anim12i 550 . . 3
8 3anass 940 . . 3
97, 8sylibr 204 . 2
10 eqid 2404 . . 3
11 breq1 4175 . . . . 5
1211anbi1d 686 . . . 4
13 breq1 4175 . . . 4
1412, 13imbi12d 312 . . 3
15 breq2 4176 . . . . 5
16 breq1 4175 . . . . 5
1715, 16anbi12d 692 . . . 4
1817imbi1d 309 . . 3
19 breq2 4176 . . . . 5
2019anbi2d 685 . . . 4
21 breq2 4176 . . . 4
2220, 21imbi12d 312 . . 3
231ecopoveq 6964 . . . . . . . 8
24233adant3 977 . . . . . . 7
251ecopoveq 6964 . . . . . . . 8
26253adant1 975 . . . . . . 7
2724, 26anbi12d 692 . . . . . 6
28 oveq12 6049 . . . . . . 7
29 vex 2919 . . . . . . . 8
30 vex 2919 . . . . . . . 8
31 vex 2919 . . . . . . . 8
32 ecopopr.com . . . . . . . 8
33 ecopopr.ass . . . . . . . 8
34 vex 2919 . . . . . . . 8
3529, 30, 31, 32, 33, 34caov411 6238 . . . . . . 7
36 vex 2919 . . . . . . . . 9
37 vex 2919 . . . . . . . . 9
3836, 30, 29, 32, 33, 37caov411 6238 . . . . . . . 8
3936, 30, 29, 32, 33, 37caov4 6237 . . . . . . . 8
4038, 39eqtr3i 2426 . . . . . . 7
4128, 35, 403eqtr4g 2461 . . . . . 6
4227, 41syl6bi 220 . . . . 5
43 ecopopr.cl . . . . . . . . . . 11
4443caovcl 6200 . . . . . . . . . 10
4543caovcl 6200 . . . . . . . . . 10
46 ovex 6065 . . . . . . . . . . 11
47 ecopopr.can . . . . . . . . . . 11
4846, 47caovcan 6210 . . . . . . . . . 10
4944, 45, 48syl2an 464 . . . . . . . . 9
50493impb 1149 . . . . . . . 8
51503com12 1157 . . . . . . 7
52513adant3l 1180 . . . . . 6
53523adant1r 1177 . . . . 5
5442, 53syld 42 . . . 4
551ecopoveq 6964 . . . . 5
56553adant2 976 . . . 4
5754, 56sylibrd 226 . . 3
5810, 14, 18, 22, 573optocl 4913 . 2
599, 58mpcom 34 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1547   wceq 1649   wcel 1721  cop 3777   class class class wbr 4172  copab 4225   cxp 4835  (class class class)co 6040 This theorem is referenced by:  ecopover  6967 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-iota 5377  df-fv 5421  df-ov 6043
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