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Theorem ecopovtrn 7433
 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:   ,,,,,,   ,S,,,,,

Proof of Theorem ecopovtrn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecopopr.1 . . . . . . 7
2 opabssxp 5079 . . . . . . 7
31, 2eqsstri 3533 . . . . . 6
43brel 5053 . . . . 5
54simpld 459 . . . 4
63brel 5053 . . . 4
75, 6anim12i 566 . . 3
8 3anass 977 . . 3
97, 8sylibr 212 . 2
10 eqid 2457 . . 3
11 breq1 4455 . . . . 5
1211anbi1d 704 . . . 4
13 breq1 4455 . . . 4
1412, 13imbi12d 320 . . 3
15 breq2 4456 . . . . 5
16 breq1 4455 . . . . 5
1715, 16anbi12d 710 . . . 4
1817imbi1d 317 . . 3
19 breq2 4456 . . . . 5
2019anbi2d 703 . . . 4
21 breq2 4456 . . . 4
2220, 21imbi12d 320 . . 3
231ecopoveq 7431 . . . . . . . 8
24233adant3 1016 . . . . . . 7
251ecopoveq 7431 . . . . . . . 8
26253adant1 1014 . . . . . . 7
2724, 26anbi12d 710 . . . . . 6
28 oveq12 6305 . . . . . . 7
29 vex 3112 . . . . . . . 8
30 vex 3112 . . . . . . . 8
31 vex 3112 . . . . . . . 8
32 ecopopr.com . . . . . . . 8
33 ecopopr.ass . . . . . . . 8
34 vex 3112 . . . . . . . 8
3529, 30, 31, 32, 33, 34caov411 6507 . . . . . . 7
36 vex 3112 . . . . . . . . 9
37 vex 3112 . . . . . . . . 9
3836, 30, 29, 32, 33, 37caov411 6507 . . . . . . . 8
3936, 30, 29, 32, 33, 37caov4 6506 . . . . . . . 8
4038, 39eqtr3i 2488 . . . . . . 7
4128, 35, 403eqtr4g 2523 . . . . . 6
4227, 41syl6bi 228 . . . . 5
43 ecopopr.cl . . . . . . . . . . 11
4443caovcl 6469 . . . . . . . . . 10
4543caovcl 6469 . . . . . . . . . 10
46 ovex 6324 . . . . . . . . . . 11
47 ecopopr.can . . . . . . . . . . 11
4846, 47caovcan 6479 . . . . . . . . . 10
4944, 45, 48syl2an 477 . . . . . . . . 9
50493impb 1192 . . . . . . . 8
51503com12 1200 . . . . . . 7
52513adant3l 1224 . . . . . 6
53523adant1r 1221 . . . . 5
5442, 53syld 44 . . . 4
551ecopoveq 7431 . . . . 5
56553adant2 1015 . . . 4
5754, 56sylibrd 234 . . 3
5810, 14, 18, 22, 573optocl 5083 . 2
599, 58mpcom 36 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035   class class class wbr 4452  {copab 4509  X.cxp 5002  (class class class)co 6296 This theorem is referenced by:  ecopover  7434 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  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-sbc 3328  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-iota 5556  df-fv 5601  df-ov 6299
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