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Theorem phtpycc 18969
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 18862 . . . 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 18960 . . . 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 3309 . . 3  |-  ( ph  ->  K  e.  ( F ( II Htpy  J ) G ) )
106, 2phtpyhtpy 18960 . . . 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 3309 . . 3  |-  ( ph  ->  L  e.  ( G ( II Htpy  J ) H ) )
133, 5, 1, 6, 2, 9, 12htpycc 18958 . 2  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
14 0elunit 10971 . . . 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 4182 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
18 simpl 444 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
1916oveq2d 6056 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
2018, 19oveq12d 6058 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 0 K ( 2  x.  s ) ) )
2119oveq1d 6055 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
2218, 21oveq12d 6058 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 0 L ( ( 2  x.  s )  -  1 ) ) )
2317, 20, 22ifbieq12d 3721 . . . . 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 6065 . . . . . 6  |-  ( 0 K ( 2  x.  s ) )  e. 
_V
25 ovex 6065 . . . . . 6  |-  ( 0 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
2624, 25ifex 3757 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
2723, 3, 26ovmpt2a 6163 . . . 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 2410 . . . 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 2410 . . . 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 18913 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
33 iihalf1 18909 . . . . . . . 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 18962 . . . . . 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 18914 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
41 iihalf2 18911 . . . . . . . . 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 18962 . . . . . . 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 18963 . . . . . . 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 2439 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 0 ) )
5129, 30, 38, 50ifbothda 3729 . . 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 2436 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
53 1elunit 10972 . . . 4  |-  1  e.  ( 0 [,] 1
)
54 simpr 448 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
5554breq1d 4182 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
56 simpl 444 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
5754oveq2d 6056 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
5856, 57oveq12d 6058 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 1 K ( 2  x.  s ) ) )
5957oveq1d 6055 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
6056, 59oveq12d 6058 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 1 L ( ( 2  x.  s )  -  1 ) ) )
6155, 58, 60ifbieq12d 3721 . . . . 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 6065 . . . . . 6  |-  ( 1 K ( 2  x.  s ) )  e. 
_V
63 ovex 6065 . . . . . 6  |-  ( 1 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
6462, 63ifex 3757 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
6561, 3, 64ovmpt2a 6163 . . . 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 2410 . . . 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 2410 . . . 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 2439 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 1 ) )
7367, 68, 69, 72ifbothda 3729 . . 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 2436 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  ( F ` 
1 ) )
751, 2, 13, 52, 74isphtpyd 18964 1  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ifcif 3699   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   0cc0 8946   1c1 8947    x. cmul 8951    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   [,]cicc 10875  TopOnctopon 16914    Cn ccn 17242   IIcii 18858   Htpy chtpy 18945   PHtpycphtpy 18946
This theorem is referenced by:  phtpcer  18973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-cn 17245  df-cnp 17246  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305  df-ii 18860  df-htpy 18948  df-phtpy 18949
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