Theorem hmphtr 14885

Description: "Is homeomorph to" is transitive.

Assertion

Ref	Expression
hmphtr	|- ((J e. Top /\ K e. Top /\ L e. Top) -> ((J ~= K /\ K ~= L) -> J ~= L))

Proof of Theorem hmphtr

Step	Hyp	Ref	Expression
1		coexg 4429	. . . . . . . . . 10 |- ((g e. (K Homeo L) /\ f e. (J Homeo K)) -> (g o. f) e. _V)
2	1	adantl 424	. . . . . . . . 9 |- (((J e. Top /\ K e. Top /\ L e. Top) /\ (g e. (K Homeo L) /\ f e. (J Homeo K))) -> (g o. f) e. _V)
3		cmphmp 14878	. . . . . . . . . . 11 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((f e. (J Homeo K) /\ g e. (K Homeo L)) -> (g o. f) e. (J Homeo L)))
4	3	ancomsd 485	. . . . . . . . . 10 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((g e. (K Homeo L) /\ f e. (J Homeo K)) -> (g o. f) e. (J Homeo L)))
5	4	imp 377	. . . . . . . . 9 |- (((J e. Top /\ K e. Top /\ L e. Top) /\ (g e. (K Homeo L) /\ f e. (J Homeo K))) -> (g o. f) e. (J Homeo L))
6		eleq1 1957	. . . . . . . . . 10 |- (h = (g o. f) -> (h e. (J Homeo L) <-> (g o. f) e. (J Homeo L)))
7	6	cla4egv 2365	. . . . . . . . 9 |- ((g o. f) e. _V -> ((g o. f) e. (J Homeo L) -> E.h h e. (J Homeo L)))
8	2, 5, 7	sylc 83	. . . . . . . 8 |- (((J e. Top /\ K e. Top /\ L e. Top) /\ (g e. (K Homeo L) /\ f e. (J Homeo K))) -> E.h h e. (J Homeo L))
9	8	exp32 408	. . . . . . 7 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (g e. (K Homeo L) -> (f e. (J Homeo K) -> E.h h e. (J Homeo L))))
10	9	19.23adv 1584	. . . . . 6 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (E.g g e. (K Homeo L) -> (f e. (J Homeo K) -> E.h h e. (J Homeo L))))
11	10	com23 36	. . . . 5 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (f e. (J Homeo K) -> (E.g g e. (K Homeo L) -> E.h h e. (J Homeo L))))
12	11	19.23adv 1584	. . . 4 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (E.f f e. (J Homeo K) -> (E.g g e. (K Homeo L) -> E.h h e. (J Homeo L))))
13	12	imp3a 388	. . 3 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((E.f f e. (J Homeo K) /\ E.g g e. (K Homeo L)) -> E.h h e. (J Homeo L)))
14		hmph 10241	. . . 4 |- ((J e. Top /\ L e. Top) -> (J ~= L <-> E.h h e. (J Homeo L)))
15	14	3adant2 895	. . 3 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (J ~= L <-> E.h h e. (J Homeo L)))
16	13, 15	sylibrd 221	. 2 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((E.f f e. (J Homeo K) /\ E.g g e. (K Homeo L)) -> J ~= L))
17		hmph 10241	. . . 4 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))
18	17	biimpd 170	. . 3 |- ((J e. Top /\ K e. Top) -> (J ~= K -> E.f f e. (J Homeo K)))
19	18	3adant3 896	. 2 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (J ~= K -> E.f f e. (J Homeo K)))
20		hmph 10241	. . . 4 |- ((K e. Top /\ L e. Top) -> (K ~= L <-> E.g g e. (K Homeo L)))
21	20	biimpd 170	. . 3 |- ((K e. Top /\ L e. Top) -> (K ~= L -> E.g g e. (K Homeo L)))
22	21	3adant1 894	. 2 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (K ~= L -> E.g g e. (K Homeo L)))
23	16, 19, 22	syl2and 508	1 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((J ~= K /\ K ~= L) -> J ~= L))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   e. wcel 1300  E.wex 1326  _Vcvv 2292   class class class wbr 3338   o. ccom 3990  (class class class)co 4884  Topctop 8857   Homeo chomeosm 10230   ~= chomeo 10231

This theorem is referenced by:  hmpher 14890

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-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-homeo 10232  df-hmph 10233

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