Theorem hmeoco 12916

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 12905	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		hmeocn 12905	. . 3 |- ( G e. ( K Homeo L ) -> G e. ( K Cn L ) )
3		cnco 12821	. . 3 |- ( ( F e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. F ) e. ( J Cn L ) )
4	1, 2, 3	syl2an 287	. 2 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Cn L ) )
5		cnvco 4788	. . 3 |- `' ( G o. F ) = ( `' F o. `' G )
6		hmeocnvcn 12906	. . . 4 |- ( G e. ( K Homeo L ) -> `' G e. ( L Cn K ) )
7		hmeocnvcn 12906	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
8		cnco 12821	. . . 4 |- ( ( `' G e. ( L Cn K ) /\ `' F e. ( K Cn J ) ) -> ( `' F o. `' G ) e. ( L Cn J ) )
9	6, 7, 8	syl2anr 288	. . 3 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( `' F o. `' G ) e. ( L Cn J ) )
10	5, 9	eqeltrid 2252	. 2 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> `' ( G o. F ) e. ( L Cn J ) )
11		ishmeo 12904	. 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 414	1 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> ( G o. F ) e. ( J Homeo L ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   `'ccnv 4602   o. ccom 4607  (class class class)co 5841   Cn ccn 12785   Homeochmeo 12900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-ral 2448  df-rex 2449  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-fv 5195  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-map 6612  df-top 12596  df-topon 12609  df-cn 12788  df-hmeo 12901

This theorem is referenced by: (None)