Theorem hmphtr 10531

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		eleq1 1534	. . . . . . . . . 10 |- (h = (g o. f) -> (h e. (J Homeo L) <-> (g o. f) e. (J Homeo L)))
2	1	cla4egv 1863	. . . . . . . . 9 |- ((g o. f) e. V -> ((g o. f) e. (J Homeo L) -> E.h h e. (J Homeo L)))
3		coexg 3524	. . . . . . . . . 10 |- ((g e. (K Homeo L) /\ f e. (J Homeo K)) -> (g o. f) e. V)
4	3	adantl 388	. . . . . . . . 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)
5		cmphmp 10521	. . . . . . . . . . 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)))
6	5	ancomsd 437	. . . . . . . . . 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)))
7	6	imp 350	. . . . . . . . 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))
8	2, 4, 7	sylc 68	. . . . . . . 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 377	. . . . . . 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 1214	. . . . . 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 32	. . . . 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 1214	. . . 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 361	. . 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 10524	. . . 4 |- ((J e. Top /\ L e. Top) -> (J ~= L <-> E.h h e. (J Homeo L)))
15	14	3adant2 798	. . 3 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (J ~= L <-> E.h h e. (J Homeo L)))
16	13, 15	sylibrd 204	. 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 10524	. . . 4 |- ((J e. Top /\ K e. Top) -> (J ~= K <-> E.f f e. (J Homeo K)))
18	17	biimpd 153	. . 3 |- ((J e. Top /\ K e. Top) -> (J ~= K -> E.f f e. (J Homeo K)))
19	18	3adant3 799	. 2 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (J ~= K -> E.f f e. (J Homeo K)))
20		hmph 10524	. . . 4 |- ((K e. Top /\ L e. Top) -> (K ~= L <-> E.g g e. (K Homeo L)))
21	20	biimpd 153	. . 3 |- ((K e. Top /\ L e. Top) -> (K ~= L -> E.g g e. (K Homeo L)))
22	21	3adant1 797	. 2 |- ((J e. Top /\ K e. Top /\ L e. Top) -> (K ~= L -> E.g g e. (K Homeo L)))
23	16, 19, 22	syl2and 459	1 |- ((J e. Top /\ K e. Top /\ L e. Top) -> ((J ~= K /\ K ~= L) -> J ~= L))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   e. wcel 958  E.wex 980  Vcvv 1811   class class class wbr 2619   o. ccom 3174  (class class class)co 3963  Topctop 7588   Homeo chomeosm 10513   ~= chomeo 10514

This theorem is referenced by:  hmpher 10536

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-homeo 10515  df-hmph 10523

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