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.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . . . 7
2		opabssxp 5079	. . . . . . 7
3	1, 2	eqsstri 3533	. . . . . 6
4	3	brel 5053	. . . . 5
5	4	simpld 459	. . . 4
6	3	brel 5053	. . . 4
7	5, 6	anim12i 566	. . 3
8		3anass 977	. . 3
9	7, 8	sylibr 212	. 2
10		eqid 2457	. . 3
11		breq1 4455	. . . . 5
12	11	anbi1d 704	. . . 4
13		breq1 4455	. . . 4
14	12, 13	imbi12d 320	. . 3
15		breq2 4456	. . . . 5
16		breq1 4455	. . . . 5
17	15, 16	anbi12d 710	. . . 4
18	17	imbi1d 317	. . 3
19		breq2 4456	. . . . 5
20	19	anbi2d 703	. . . 4
21		breq2 4456	. . . 4
22	20, 21	imbi12d 320	. . 3
23	1	ecopoveq 7431	. . . . . . . 8
24	23	3adant3 1016	. . . . . . 7
25	1	ecopoveq 7431	. . . . . . . 8
26	25	3adant1 1014	. . . . . . 7
27	24, 26	anbi12d 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
35	29, 30, 31, 32, 33, 34	caov411 6507	. . . . . . 7
36		vex 3112	. . . . . . . . 9
37		vex 3112	. . . . . . . . 9
38	36, 30, 29, 32, 33, 37	caov411 6507	. . . . . . . 8
39	36, 30, 29, 32, 33, 37	caov4 6506	. . . . . . . 8
40	38, 39	eqtr3i 2488	. . . . . . 7
41	28, 35, 40	3eqtr4g 2523	. . . . . 6
42	27, 41	syl6bi 228	. . . . 5
43		ecopopr.cl	. . . . . . . . . . 11
44	43	caovcl 6469	. . . . . . . . . 10
45	43	caovcl 6469	. . . . . . . . . 10
46		ovex 6324	. . . . . . . . . . 11
47		ecopopr.can	. . . . . . . . . . 11
48	46, 47	caovcan 6479	. . . . . . . . . 10
49	44, 45, 48	syl2an 477	. . . . . . . . 9
50	49	3impb 1192	. . . . . . . 8
51	50	3com12 1200	. . . . . . 7
52	51	3adant3l 1224	. . . . . 6
53	52	3adant1r 1221	. . . . 5
54	42, 53	syld 44	. . . 4
55	1	ecopoveq 7431	. . . . 5
56	55	3adant2 1015	. . . 4
57	54, 56	sylibrd 234	. . 3
58	10, 14, 18, 22, 57	3optocl 5083	. 2
59	9, 58	mpcom 36	1

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