Theorem hmeoco 15307

Description: The composite of two homeomorphisms is a homeomorphism. (Contributed by FL, 9-Mar-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
hmeoco	|- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Homeo L ) )

Proof of Theorem hmeoco

Step	Hyp	Ref	Expression
1		hmeocn 15296	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		hmeocn 15296	. . 3 |- ( G e. ( K Homeo L ) -> G e. ( K Cn L ) )
3		cnco 15212	. . 3 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )
4	1, 2, 3	syl2an 289	. 2 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Cn L ) )
5		cnvco 4945	. . 3 |- `' ( G o. F ) = ( `' F o. `' G )
6		hmeocnvcn 15297	. . . 4 |- ( G e. ( K Homeo L ) -> `' G e. ( L Cn K ) )
7		hmeocnvcn 15297	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
8		cnco 15212	. . . 4 |- ( ( `' G e. ( L Cn K ) /\ `' F e. ( K Cn J ) ) -> ( `' F o. `' G ) e. ( L Cn J ) )
9	6, 7, 8	syl2anr 290	. . 3 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( `' F o. `' G ) e. ( L Cn J ) )
10	5, 9	eqeltrid 2321	. 2 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> `' ( G o. F ) e. ( L Cn J ) )
11		ishmeo 15295	. 2 |- ( ( G o. F ) e. ( J Homeo L ) <-> ( ( G o. F ) e. ( J Cn L ) /\ `' ( G o. F ) e. ( L Cn J ) ) )
12	4, 10, 11	sylanbrc 417	1 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Homeo L ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2205   `'ccnv 4753   o. ccom 4758  (class class class)co 6058   Cn ccn 15176   Homeochmeo 15291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-top 14989  df-topon 15002  df-cn 15179  df-hmeo 15292

This theorem is referenced by: (None)