Theorem hmeoco 14636

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 14625	. . 3 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2		hmeocn 14625	. . 3 |- ( G e. ( K Homeo L ) -> G e. ( K Cn L ) )
3		cnco 14541	. . 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 4852	. . 3 |- `' ( G o. F ) = ( `' F o. `' G )
6		hmeocnvcn 14626	. . . 4 |- ( G e. ( K Homeo L ) -> `' G e. ( L Cn K ) )
7		hmeocnvcn 14626	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
8		cnco 14541	. . . 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 2283	. 2 |- ( ( F e. ( J Homeo K ) /\ G e. ( K Homeo L ) ) -> `' ( G o. F ) e. ( L Cn J ) )
11		ishmeo 14624	. 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 2167   `'ccnv 4663   o. ccom 4668  (class class class)co 5925   Cn ccn 14505   Homeochmeo 14620

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-top 14318  df-topon 14331  df-cn 14508  df-hmeo 14621

This theorem is referenced by: (None)