Theorem dirtr 10356

Description: A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.)

Assertion

Ref	Expression
dirtr	|- (((D e. Dir /\ C e. H) /\ (ADB /\ BDC)) -> ADC)

Proof of Theorem dirtr

Step	Hyp	Ref	Expression
1		brrelex 4028	. . . 4 |- ((Rel D /\ ADB) -> A e. _V)
2		reldir 10353	. . . 4 |- (D e. Dir -> Rel D)
3	1, 2	sylan 497	. . 3 |- ((D e. Dir /\ ADB) -> A e. _V)
4	3	ad2ant2r 445	. 2 |- (((D e. Dir /\ C e. H) /\ (ADB /\ BDC)) -> A e. _V)
5		brrelex 4028	. . . 4 |- ((Rel D /\ BDC) -> B e. _V)
6	5, 2	sylan 497	. . 3 |- ((D e. Dir /\ BDC) -> B e. _V)
7	6	ad2ant2rl 447	. 2 |- (((D e. Dir /\ C e. H) /\ (ADB /\ BDC)) -> B e. _V)
8		eqid 1884	. . . . . . . . . . . . . 14 |- U.U.D = U.U.D
9	8	isdir 10352	. . . . . . . . . . . . 13 |- (D e. Dir -> (D e. Dir <-> ((Rel D /\ ( _I |` U.U.D) C_ D) /\ ((D o. D) C_ D /\ A.x e. U.U.DA.y e. U.U.DE.z e. U.U.D(xDz /\ yDz)))))
10	9	ibi 652	. . . . . . . . . . . 12 |- (D e. Dir -> ((Rel D /\ ( _I |` U.U.D) C_ D) /\ ((D o. D) C_ D /\ A.x e. U.U.DA.y e. U.U.DE.z e. U.U.D(xDz /\ yDz))))
11	10	simprd 352	. . . . . . . . . . 11 |- (D e. Dir -> ((D o. D) C_ D /\ A.x e. U.U.DA.y e. U.U.DE.z e. U.U.D(xDz /\ yDz)))
12	11	simplld 348	. . . . . . . . . 10 |- (D e. Dir -> (D o. D) C_ D)
13	12	ad2antrr 440	. . . . . . . . 9 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> (D o. D) C_ D)
14		cotr 4302	. . . . . . . . 9 |- ((D o. D) C_ D <-> A.xA.yA.z((xDy /\ yDz) -> xDz))
15	13, 14	sylib 215	. . . . . . . 8 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> A.xA.yA.z((xDy /\ yDz) -> xDz))
16		breq1 3341	. . . . . . . . . . . . 13 |- (x = A -> (xDy <-> ADy))
17	16	anbi1d 679	. . . . . . . . . . . 12 |- (x = A -> ((xDy /\ yDz) <-> (ADy /\ yDz)))
18		breq1 3341	. . . . . . . . . . . 12 |- (x = A -> (xDz <-> ADz))
19	17, 18	imbi12d 688	. . . . . . . . . . 11 |- (x = A -> (((xDy /\ yDz) -> xDz) <-> ((ADy /\ yDz) -> ADz)))
20	19	2albidv 1658	. . . . . . . . . 10 |- (x = A -> (A.yA.z((xDy /\ yDz) -> xDz) <-> A.yA.z((ADy /\ yDz) -> ADz)))
21	20	cla4gv 2364	. . . . . . . . 9 |- (A e. _V -> (A.xA.yA.z((xDy /\ yDz) -> xDz) -> A.yA.z((ADy /\ yDz) -> ADz)))
22	21	ad2antrl 442	. . . . . . . 8 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> (A.xA.yA.z((xDy /\ yDz) -> xDz) -> A.yA.z((ADy /\ yDz) -> ADz)))
23	15, 22	mpd 29	. . . . . . 7 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> A.yA.z((ADy /\ yDz) -> ADz))
24		breq2 3342	. . . . . . . . . . . 12 |- (y = B -> (ADy <-> ADB))
25		breq1 3341	. . . . . . . . . . . 12 |- (y = B -> (yDz <-> BDz))
26	24, 25	anbi12d 690	. . . . . . . . . . 11 |- (y = B -> ((ADy /\ yDz) <-> (ADB /\ BDz)))
27	26	imbi1d 675	. . . . . . . . . 10 |- (y = B -> (((ADy /\ yDz) -> ADz) <-> ((ADB /\ BDz) -> ADz)))
28	27	albidv 1656	. . . . . . . . 9 |- (y = B -> (A.z((ADy /\ yDz) -> ADz) <-> A.z((ADB /\ BDz) -> ADz)))
29	28	cla4gv 2364	. . . . . . . 8 |- (B e. _V -> (A.yA.z((ADy /\ yDz) -> ADz) -> A.z((ADB /\ BDz) -> ADz)))
30	29	ad2antll 443	. . . . . . 7 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> (A.yA.z((ADy /\ yDz) -> ADz) -> A.z((ADB /\ BDz) -> ADz)))
31	23, 30	mpd 29	. . . . . 6 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> A.z((ADB /\ BDz) -> ADz))
32		breq2 3342	. . . . . . . . . 10 |- (z = C -> (BDz <-> BDC))
33	32	anbi2d 678	. . . . . . . . 9 |- (z = C -> ((ADB /\ BDz) <-> (ADB /\ BDC)))
34		breq2 3342	. . . . . . . . 9 |- (z = C -> (ADz <-> ADC))
35	33, 34	imbi12d 688	. . . . . . . 8 |- (z = C -> (((ADB /\ BDz) -> ADz) <-> ((ADB /\ BDC) -> ADC)))
36	35	cla4gv 2364	. . . . . . 7 |- (C e. H -> (A.z((ADB /\ BDz) -> ADz) -> ((ADB /\ BDC) -> ADC)))
37	36	ad2antlr 441	. . . . . 6 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> (A.z((ADB /\ BDz) -> ADz) -> ((ADB /\ BDC) -> ADC)))
38	31, 37	mpd 29	. . . . 5 |- (((D e. Dir /\ C e. H) /\ (A e. _V /\ B e. _V)) -> ((ADB /\ BDC) -> ADC))
39	38	ex 402	. . . 4 |- ((D e. Dir /\ C e. H) -> ((A e. _V /\ B e. _V) -> ((ADB /\ BDC) -> ADC)))
40	39	com23 36	. . 3 |- ((D e. Dir /\ C e. H) -> ((ADB /\ BDC) -> ((A e. _V /\ B e. _V) -> ADC)))
41	40	imp 377	. 2 |- (((D e. Dir /\ C e. H) /\ (ADB /\ BDC)) -> ((A e. _V /\ B e. _V) -> ADC))
42	4, 7, 41	mp2and 767	1 |- (((D e. Dir /\ C e. H) /\ (ADB /\ BDC)) -> ADC)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  U.cuni 3177   class class class wbr 3338   _I cid 3582   |` cres 3988   o. ccom 3990  Rel wrel 3991  Dircdir 10348

This theorem is referenced by:  tailfb 15639

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-ral 2109  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-rel 4001  df-co 4003  df-res 4006  df-dir 10350

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