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Theorem phtpycc 19018
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 18911 . . . 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 19009 . . . 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 3351 . . 3  |-  ( ph  ->  K  e.  ( F ( II Htpy  J ) G ) )
106, 2phtpyhtpy 19009 . . . 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 3351 . . 3  |-  ( ph  ->  L  e.  ( G ( II Htpy  J ) H ) )
133, 5, 1, 6, 2, 9, 12htpycc 19007 . 2  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
14 0elunit 11017 . . . 4  |-  0  e.  ( 0 [,] 1
)
15 simpr 449 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
16 simpr 449 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
1716breq1d 4224 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
18 simpl 445 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
1916oveq2d 6099 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
2018, 19oveq12d 6101 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 0 K ( 2  x.  s ) ) )
2119oveq1d 6098 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
2218, 21oveq12d 6101 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 0 L ( ( 2  x.  s )  -  1 ) ) )
2317, 20, 22ifbieq12d 3763 . . . . 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 6108 . . . . . 6  |-  ( 0 K ( 2  x.  s ) )  e. 
_V
25 ovex 6108 . . . . . 6  |-  ( 0 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
2624, 25ifex 3799 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
2723, 3, 26ovmpt2a 6206 . . . 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 646 . . 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 2444 . . . 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 2444 . . . 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 732 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  ph )
32 elii1 18962 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
33 iihalf1 18958 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
3432, 33sylbir 206 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  /\  s  <_  ( 1  / 
2 ) )  -> 
( 2  x.  s
)  e.  ( 0 [,] 1 ) )
3534adantll 696 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
361, 6, 8phtpyi 19011 . . . . . 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 644 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3837simpld 447 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
0 K ( 2  x.  s ) )  =  ( F ` 
0 ) )
39 simpll 732 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  ph )
40 elii2 18963 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
41 iihalf2 18960 . . . . . . . . 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 696 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 2  x.  s )  -  1 )  e.  ( 0 [,] 1 ) )
446, 2, 11phtpyi 19011 . . . . . . 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 644 . . . . . 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 447 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 0 ) )
471, 6, 8phtpy01 19012 . . . . . . 7  |-  ( ph  ->  ( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4847ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4948simpld 447 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  0
)  =  ( G `
 0 ) )
5046, 49eqtr4d 2473 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 0 ) )
5129, 30, 38, 50ifbothda 3771 . . 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 2470 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
53 1elunit 11018 . . . 4  |-  1  e.  ( 0 [,] 1
)
54 simpr 449 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
5554breq1d 4224 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
56 simpl 445 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
5754oveq2d 6099 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
5856, 57oveq12d 6101 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 1 K ( 2  x.  s ) ) )
5957oveq1d 6098 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
6056, 59oveq12d 6101 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 1 L ( ( 2  x.  s )  -  1 ) ) )
6155, 58, 60ifbieq12d 3763 . . . . 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 6108 . . . . . 6  |-  ( 1 K ( 2  x.  s ) )  e. 
_V
63 ovex 6108 . . . . . 6  |-  ( 1 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
6462, 63ifex 3799 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
6561, 3, 64ovmpt2a 6206 . . . 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 646 . . 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 2444 . . . 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 2444 . . . 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 451 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
1 K ( 2  x.  s ) )  =  ( F ` 
1 ) )
7045simprd 451 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 1 ) )
7148simprd 451 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  1
)  =  ( G `
 1 ) )
7270, 71eqtr4d 2473 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 1 ) )
7367, 68, 69, 72ifbothda 3771 . . 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 2470 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  ( F ` 
1 ) )
751, 2, 13, 52, 74isphtpyd 19013 1  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ifcif 3741   class class class wbr 4214   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   0cc0 8992   1c1 8993    x. cmul 8997    <_ cle 9123    - cmin 9293    / cdiv 9679   2c2 10051   [,]cicc 10921  TopOnctopon 16961    Cn ccn 17290   IIcii 18907   Htpy chtpy 18994   PHtpycphtpy 18995
This theorem is referenced by:  phtpcer  19022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-cn 17293  df-cnp 17294  df-tx 17596  df-hmeo 17789  df-xms 18352  df-ms 18353  df-tms 18354  df-ii 18909  df-htpy 18997  df-phtpy 18998
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