Theorem ecopoprtrn 5370

Description: Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is transitive.

Hypotheses

Ref	Expression
ecopopr.1	|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}
ecopopr.com	|- (xFy) = (yFx)
ecopopr.cl	|- ((x e. S /\ y e. S) -> (xFy) e. S)
ecopopr.ass	|- ((xFy)Fz) = (xF(yFz))
ecopopr.can	|- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))
ecopopr.3	|- B e. _V
ecopopr.4	|- C e. _V

Assertion

Ref	Expression
ecopoprtrn	|- ((ARB /\ BRC) -> ARC)

Distinct variable groups:   x,y,z,w,v,u,F   x,S,y,z,w,v,u

Proof of Theorem ecopoprtrn

Step	Hyp	Ref	Expression
1		ecopopr.3	. . . . . 6 |- B e. _V
2		ecopopr.1	. . . . . . 7 |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}
3		opabssxp 4060	. . . . . . 7 |- {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))} C_ ((S X. S) X. (S X. S))
4	2, 3	eqsstri 2647	. . . . . 6 |- R C_ ((S X. S) X. (S X. S))
5	1, 4	brel 4048	. . . . 5 |- (ARB -> (A e. (S X. S) /\ B e. (S X. S)))
6	5	simplld 348	. . . 4 |- (ARB -> A e. (S X. S))
7		ecopopr.4	. . . . 5 |- C e. _V
8	7, 4	brel 4048	. . . 4 |- (BRC -> (B e. (S X. S) /\ C e. (S X. S)))
9	6, 8	anim12i 360	. . 3 |- ((ARB /\ BRC) -> (A e. (S X. S) /\ (B e. (S X. S) /\ C e. (S X. S))))
10		3anass 862	. . 3 |- ((A e. (S X. S) /\ B e. (S X. S) /\ C e. (S X. S)) <-> (A e. (S X. S) /\ (B e. (S X. S) /\ C e. (S X. S))))
11	9, 10	sylibr 217	. 2 |- ((ARB /\ BRC) -> (A e. (S X. S) /\ B e. (S X. S) /\ C e. (S X. S)))
12		eqid 1884	. . 3 |- (S X. S) = (S X. S)
13		breq1 3341	. . . . 5 |- (<.f, g>. = A -> (<.f, g>.R<.h, t>. <-> AR<.h, t>.))
14	13	anbi1d 679	. . . 4 |- (<.f, g>. = A -> ((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) <-> (AR<.h, t>. /\ <.h, t>.R<.s, r>.)))
15		breq1 3341	. . . 4 |- (<.f, g>. = A -> (<.f, g>.R<.s, r>. <-> AR<.s, r>.))
16	14, 15	imbi12d 688	. . 3 |- (<.f, g>. = A -> (((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) -> <.f, g>.R<.s, r>.) <-> ((AR<.h, t>. /\ <.h, t>.R<.s, r>.) -> AR<.s, r>.)))
17		breq2 3342	. . . . 5 |- (<.h, t>. = B -> (AR<.h, t>. <-> ARB))
18		breq1 3341	. . . . 5 |- (<.h, t>. = B -> (<.h, t>.R<.s, r>. <-> BR<.s, r>.))
19	17, 18	anbi12d 690	. . . 4 |- (<.h, t>. = B -> ((AR<.h, t>. /\ <.h, t>.R<.s, r>.) <-> (ARB /\ BR<.s, r>.)))
20	19	imbi1d 675	. . 3 |- (<.h, t>. = B -> (((AR<.h, t>. /\ <.h, t>.R<.s, r>.) -> AR<.s, r>.) <-> ((ARB /\ BR<.s, r>.) -> AR<.s, r>.)))
21		breq2 3342	. . . . 5 |- (<.s, r>. = C -> (BR<.s, r>. <-> BRC))
22	21	anbi2d 678	. . . 4 |- (<.s, r>. = C -> ((ARB /\ BR<.s, r>.) <-> (ARB /\ BRC)))
23		breq2 3342	. . . 4 |- (<.s, r>. = C -> (AR<.s, r>. <-> ARC))
24	22, 23	imbi12d 688	. . 3 |- (<.s, r>. = C -> (((ARB /\ BR<.s, r>.) -> AR<.s, r>.) <-> ((ARB /\ BRC) -> ARC)))
25	2	ecopopreq 5367	. . . . . . . 8 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> (fFt) = (gFh)))
26	25	3adant3 896	. . . . . . 7 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (<.f, g>.R<.h, t>. <-> (fFt) = (gFh)))
27	2	ecopopreq 5367	. . . . . . . 8 |- (((h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (<.h, t>.R<.s, r>. <-> (hFr) = (tFs)))
28	27	3adant1 894	. . . . . . 7 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (<.h, t>.R<.s, r>. <-> (hFr) = (tFs)))
29	26, 28	anbi12d 690	. . . . . 6 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> ((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) <-> ((fFt) = (gFh) /\ (hFr) = (tFs))))
30		opreq12 4891	. . . . . . 7 |- (((fFt) = (gFh) /\ (hFr) = (tFs)) -> ((fFt)F(hFr)) = ((gFh)F(tFs)))
31		visset 2295	. . . . . . . 8 |- h e. _V
32		visset 2295	. . . . . . . 8 |- t e. _V
33		visset 2295	. . . . . . . 8 |- f e. _V
34		ecopopr.com	. . . . . . . 8 |- (xFy) = (yFx)
35		ecopopr.ass	. . . . . . . 8 |- ((xFy)Fz) = (xF(yFz))
36		visset 2295	. . . . . . . 8 |- r e. _V
37	31, 32, 33, 34, 35, 36	caopr411 4998	. . . . . . 7 |- ((hFt)F(fFr)) = ((fFt)F(hFr))
38		visset 2295	. . . . . . . . 9 |- g e. _V
39		visset 2295	. . . . . . . . 9 |- s e. _V
40	38, 32, 31, 34, 35, 39	caopr411 4998	. . . . . . . 8 |- ((gFt)F(hFs)) = ((hFt)F(gFs))
41	38, 32, 31, 34, 35, 39	caopr4 4997	. . . . . . . 8 |- ((gFt)F(hFs)) = ((gFh)F(tFs))
42	40, 41	eqtr3i 1910	. . . . . . 7 |- ((hFt)F(gFs)) = ((gFh)F(tFs))
43	30, 37, 42	3eqtr4g 1953	. . . . . 6 |- (((fFt) = (gFh) /\ (hFr) = (tFs)) -> ((hFt)F(fFr)) = ((hFt)F(gFs)))
44	29, 43	syl6bi 231	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> ((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) -> ((hFt)F(fFr)) = ((hFt)F(gFs))))
45		oprex 4907	. . . . . . . . . . 11 |- (gFs) e. _V
46		ecopopr.can	. . . . . . . . . . 11 |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))
47	45, 46	caoprcan 4988	. . . . . . . . . 10 |- (((hFt) e. S /\ (fFr) e. S) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
48		ecopopr.cl	. . . . . . . . . . 11 |- ((x e. S /\ y e. S) -> (xFy) e. S)
49	48	caoprcl 4985	. . . . . . . . . 10 |- ((h e. S /\ t e. S) -> (hFt) e. S)
50	48	caoprcl 4985	. . . . . . . . . 10 |- ((f e. S /\ r e. S) -> (fFr) e. S)
51	47, 49, 50	syl2an 503	. . . . . . . . 9 |- (((h e. S /\ t e. S) /\ (f e. S /\ r e. S)) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
52	51	3impb 1063	. . . . . . . 8 |- (((h e. S /\ t e. S) /\ f e. S /\ r e. S) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
53	52	3com12 1071	. . . . . . 7 |- ((f e. S /\ (h e. S /\ t e. S) /\ r e. S) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
54	53	3adant3l 1094	. . . . . 6 |- ((f e. S /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
55	54	3adant1r 1091	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (((hFt)F(fFr)) = ((hFt)F(gFs)) -> (fFr) = (gFs)))
56	44, 55	syld 30	. . . 4 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> ((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) -> (fFr) = (gFs)))
57	2	ecopopreq 5367	. . . . 5 |- (((f e. S /\ g e. S) /\ (s e. S /\ r e. S)) -> (<.f, g>.R<.s, r>. <-> (fFr) = (gFs)))
58	57	3adant2 895	. . . 4 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> (<.f, g>.R<.s, r>. <-> (fFr) = (gFs)))
59	56, 58	sylibrd 221	. . 3 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S) /\ (s e. S /\ r e. S)) -> ((<.f, g>.R<.h, t>. /\ <.h, t>.R<.s, r>.) -> <.f, g>.R<.s, r>.))
60	12, 16, 20, 24, 59	3optocl 4063	. 2 |- ((A e. (S X. S) /\ B e. (S X. S) /\ C e. (S X. S)) -> ((ARB /\ BRC) -> ARC))
61	11, 60	mpcom 60	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  (class class class)co 4884

This theorem is referenced by:  ecopoprer 5371

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

Copyright terms: Public domain