MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phtpycc Structured version   Unicode version

Theorem phtpycc 19008
Description: Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
Hypotheses
Ref Expression
phtpycc.1  |-  M  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( x K ( 2  x.  y ) ) ,  ( x L ( ( 2  x.  y
)  -  1 ) ) ) )
phtpycc.3  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
phtpycc.4  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
phtpycc.5  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
phtpycc.6  |-  ( ph  ->  K  e.  ( F ( PHtpy `  J ) G ) )
phtpycc.7  |-  ( ph  ->  L  e.  ( G ( PHtpy `  J ) H ) )
Assertion
Ref Expression
phtpycc  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Distinct variable groups:    x, y, J    x, K, y    ph, x, y    x, L, y
Allowed substitution hints:    F( x, y)    G( x, y)    H( x, y)    M( x, y)

Proof of Theorem phtpycc
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 phtpycc.3 . 2  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
2 phtpycc.5 . 2  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
3 phtpycc.1 . . 3  |-  M  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 ) 
|->  if ( y  <_ 
( 1  /  2
) ,  ( x K ( 2  x.  y ) ) ,  ( x L ( ( 2  x.  y
)  -  1 ) ) ) )
4 iitopon 18901 . . . 4  |-  II  e.  (TopOn `  ( 0 [,] 1 ) )
54a1i 11 . . 3  |-  ( ph  ->  II  e.  (TopOn `  ( 0 [,] 1
) ) )
6 phtpycc.4 . . 3  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
71, 6phtpyhtpy 18999 . . . 4  |-  ( ph  ->  ( F ( PHtpy `  J ) G ) 
C_  ( F ( II Htpy  J ) G ) )
8 phtpycc.6 . . . 4  |-  ( ph  ->  K  e.  ( F ( PHtpy `  J ) G ) )
97, 8sseldd 3341 . . 3  |-  ( ph  ->  K  e.  ( F ( II Htpy  J ) G ) )
106, 2phtpyhtpy 18999 . . . 4  |-  ( ph  ->  ( G ( PHtpy `  J ) H ) 
C_  ( G ( II Htpy  J ) H ) )
11 phtpycc.7 . . . 4  |-  ( ph  ->  L  e.  ( G ( PHtpy `  J ) H ) )
1210, 11sseldd 3341 . . 3  |-  ( ph  ->  L  e.  ( G ( II Htpy  J ) H ) )
133, 5, 1, 6, 2, 9, 12htpycc 18997 . 2  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
14 0elunit 11007 . . . 4  |-  0  e.  ( 0 [,] 1
)
15 simpr 448 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
16 simpr 448 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
1716breq1d 4214 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
18 simpl 444 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
1916oveq2d 6089 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
2018, 19oveq12d 6091 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 0 K ( 2  x.  s ) ) )
2119oveq1d 6088 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
2218, 21oveq12d 6091 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 0 L ( ( 2  x.  s )  -  1 ) ) )
2317, 20, 22ifbieq12d 3753 . . . . 5  |-  ( ( x  =  0  /\  y  =  s )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x K ( 2  x.  y ) ) ,  ( x L ( ( 2  x.  y )  -  1 ) ) )  =  if ( s  <_ 
( 1  /  2
) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s
)  -  1 ) ) ) )
24 ovex 6098 . . . . . 6  |-  ( 0 K ( 2  x.  s ) )  e. 
_V
25 ovex 6098 . . . . . 6  |-  ( 0 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
2624, 25ifex 3789 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
2723, 3, 26ovmpt2a 6196 . . . 4  |-  ( ( 0  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 0 M s )  =  if ( s  <_  (
1  /  2 ) ,  ( 0 K ( 2  x.  s
) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) ) )
2814, 15, 27sylancr 645 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) ) )
29 eqeq1 2441 . . . 4  |-  ( ( 0 K ( 2  x.  s ) )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) )  -> 
( ( 0 K ( 2  x.  s
) )  =  ( F `  0 )  <-> 
if ( s  <_ 
( 1  /  2
) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s
)  -  1 ) ) )  =  ( F `  0 ) ) )
30 eqeq1 2441 . . . 4  |-  ( ( 0 L ( ( 2  x.  s )  -  1 ) )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) )  -> 
( ( 0 L ( ( 2  x.  s )  -  1 ) )  =  ( F `  0 )  <-> 
if ( s  <_ 
( 1  /  2
) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s
)  -  1 ) ) )  =  ( F `  0 ) ) )
31 simpll 731 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  ph )
32 elii1 18952 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
33 iihalf1 18948 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
3432, 33sylbir 205 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  /\  s  <_  ( 1  / 
2 ) )  -> 
( 2  x.  s
)  e.  ( 0 [,] 1 ) )
3534adantll 695 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
361, 6, 8phtpyi 19001 . . . . . 6  |-  ( (
ph  /\  ( 2  x.  s )  e.  ( 0 [,] 1
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3731, 35, 36syl2anc 643 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3837simpld 446 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
0 K ( 2  x.  s ) )  =  ( F ` 
0 ) )
39 simpll 731 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  ph )
40 elii2 18953 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
41 iihalf2 18950 . . . . . . . . 9  |-  ( s  e.  ( ( 1  /  2 ) [,] 1 )  ->  (
( 2  x.  s
)  -  1 )  e.  ( 0 [,] 1 ) )
4240, 41syl 16 . . . . . . . 8  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  ( ( 2  x.  s )  - 
1 )  e.  ( 0 [,] 1 ) )
4342adantll 695 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 2  x.  s )  -  1 )  e.  ( 0 [,] 1 ) )
446, 2, 11phtpyi 19001 . . . . . . 7  |-  ( (
ph  /\  ( (
2  x.  s )  -  1 )  e.  ( 0 [,] 1
) )  ->  (
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 0 )  /\  ( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 1 ) ) )
4539, 43, 44syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 0 L ( ( 2  x.  s )  -  1 ) )  =  ( G `  0 )  /\  ( 1 L ( ( 2  x.  s )  -  1 ) )  =  ( G `  1 ) ) )
4645simpld 446 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 0 ) )
471, 6, 8phtpy01 19002 . . . . . . 7  |-  ( ph  ->  ( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4847ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4948simpld 446 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  0
)  =  ( G `
 0 ) )
5046, 49eqtr4d 2470 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 0 ) )
5129, 30, 38, 50ifbothda 3761 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( 0 K ( 2  x.  s
) ) ,  ( 0 L ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 0 ) )
5228, 51eqtrd 2467 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
53 1elunit 11008 . . . 4  |-  1  e.  ( 0 [,] 1
)
54 simpr 448 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
5554breq1d 4214 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
56 simpl 444 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
5754oveq2d 6089 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
5856, 57oveq12d 6091 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 1 K ( 2  x.  s ) ) )
5957oveq1d 6088 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
6056, 59oveq12d 6091 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 1 L ( ( 2  x.  s )  -  1 ) ) )
6155, 58, 60ifbieq12d 3753 . . . . 5  |-  ( ( x  =  1  /\  y  =  s )  ->  if ( y  <_  ( 1  / 
2 ) ,  ( x K ( 2  x.  y ) ) ,  ( x L ( ( 2  x.  y )  -  1 ) ) )  =  if ( s  <_ 
( 1  /  2
) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s
)  -  1 ) ) ) )
62 ovex 6098 . . . . . 6  |-  ( 1 K ( 2  x.  s ) )  e. 
_V
63 ovex 6098 . . . . . 6  |-  ( 1 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
6462, 63ifex 3789 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
6561, 3, 64ovmpt2a 6196 . . . 4  |-  ( ( 1  e.  ( 0 [,] 1 )  /\  s  e.  ( 0 [,] 1 ) )  ->  ( 1 M s )  =  if ( s  <_  (
1  /  2 ) ,  ( 1 K ( 2  x.  s
) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) ) )
6653, 15, 65sylancr 645 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) ) )
67 eqeq1 2441 . . . 4  |-  ( ( 1 K ( 2  x.  s ) )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) )  -> 
( ( 1 K ( 2  x.  s
) )  =  ( F `  1 )  <-> 
if ( s  <_ 
( 1  /  2
) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s
)  -  1 ) ) )  =  ( F `  1 ) ) )
68 eqeq1 2441 . . . 4  |-  ( ( 1 L ( ( 2  x.  s )  -  1 ) )  =  if ( s  <_  ( 1  / 
2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) )  -> 
( ( 1 L ( ( 2  x.  s )  -  1 ) )  =  ( F `  1 )  <-> 
if ( s  <_ 
( 1  /  2
) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s
)  -  1 ) ) )  =  ( F `  1 ) ) )
6937simprd 450 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
1 K ( 2  x.  s ) )  =  ( F ` 
1 ) )
7045simprd 450 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 1 ) )
7148simprd 450 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  1
)  =  ( G `
 1 ) )
7270, 71eqtr4d 2470 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 1 ) )
7367, 68, 69, 72ifbothda 3761 . . 3  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  if ( s  <_  (
1  /  2 ) ,  ( 1 K ( 2  x.  s
) ) ,  ( 1 L ( ( 2  x.  s )  -  1 ) ) )  =  ( F `
 1 ) )
7466, 73eqtrd 2467 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  ( F ` 
1 ) )
751, 2, 13, 52, 74isphtpyd 19003 1  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   0cc0 8982   1c1 8983    x. cmul 8987    <_ cle 9113    - cmin 9283    / cdiv 9669   2c2 10041   [,]cicc 10911  TopOnctopon 16951    Cn ccn 17280   IIcii 18897   Htpy chtpy 18984   PHtpycphtpy 18985
This theorem is referenced by:  phtpcer  19012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-cn 17283  df-cnp 17284  df-tx 17586  df-hmeo 17779  df-xms 18342  df-ms 18343  df-tms 18344  df-ii 18899  df-htpy 18987  df-phtpy 18988
  Copyright terms: Public domain W3C validator