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Theorem phtpycc 20575
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 20467 . . . 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 20566 . . . 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 3369 . . 3  |-  ( ph  ->  K  e.  ( F ( II Htpy  J ) G ) )
106, 2phtpyhtpy 20566 . . . 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 3369 . . 3  |-  ( ph  ->  L  e.  ( G ( II Htpy  J ) H ) )
133, 5, 1, 6, 2, 9, 12htpycc 20564 . 2  |-  ( ph  ->  M  e.  ( F ( II Htpy  J ) H ) )
14 0elunit 11415 . . . 4  |-  0  e.  ( 0 [,] 1
)
15 simpr 461 . . . 4  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  s  e.  ( 0 [,] 1
) )
16 simpr 461 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  y  =  s )
1716breq1d 4314 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
18 simpl 457 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  x  =  0 )
1916oveq2d 6119 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
2018, 19oveq12d 6121 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 0 K ( 2  x.  s ) ) )
2119oveq1d 6118 . . . . . . 7  |-  ( ( x  =  0  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
2218, 21oveq12d 6121 . . . . . 6  |-  ( ( x  =  0  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 0 L ( ( 2  x.  s )  -  1 ) ) )
2317, 20, 22ifbieq12d 3828 . . . . 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 6128 . . . . . 6  |-  ( 0 K ( 2  x.  s ) )  e. 
_V
25 ovex 6128 . . . . . 6  |-  ( 0 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
2624, 25ifex 3870 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 0 K ( 2  x.  s ) ) ,  ( 0 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
2723, 3, 26ovmpt2a 6233 . . . 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 663 . . 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 2449 . . . 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 2449 . . . 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 753 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  ph )
32 elii1 20519 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  <->  ( s  e.  ( 0 [,] 1
)  /\  s  <_  ( 1  /  2 ) ) )
33 iihalf1 20515 . . . . . . . 8  |-  ( s  e.  ( 0 [,] ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
3432, 33sylbir 213 . . . . . . 7  |-  ( ( s  e.  ( 0 [,] 1 )  /\  s  <_  ( 1  / 
2 ) )  -> 
( 2  x.  s
)  e.  ( 0 [,] 1 ) )
3534adantll 713 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
2  x.  s )  e.  ( 0 [,] 1 ) )
361, 6, 8phtpyi 20568 . . . . . 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 661 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
( 0 K ( 2  x.  s ) )  =  ( F `
 0 )  /\  ( 1 K ( 2  x.  s ) )  =  ( F `
 1 ) ) )
3837simpld 459 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
0 K ( 2  x.  s ) )  =  ( F ` 
0 ) )
39 simpll 753 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  ->  ph )
40 elii2 20520 . . . . . . . . 9  |-  ( ( s  e.  ( 0 [,] 1 )  /\  -.  s  <_  ( 1  /  2 ) )  ->  s  e.  ( ( 1  /  2
) [,] 1 ) )
41 iihalf2 20517 . . . . . . . . 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 713 . . . . . . 7  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( 2  x.  s )  -  1 )  e.  ( 0 [,] 1 ) )
446, 2, 11phtpyi 20568 . . . . . . 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 661 . . . . . 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 459 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 0 ) )
471, 6, 8phtpy01 20569 . . . . . . 7  |-  ( ph  ->  ( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4847ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( ( F ` 
0 )  =  ( G `  0 )  /\  ( F ` 
1 )  =  ( G `  1 ) ) )
4948simpld 459 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  0
)  =  ( G `
 0 ) )
5046, 49eqtr4d 2478 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 0 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 0 ) )
5129, 30, 38, 50ifbothda 3836 . . 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 2475 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
0 M s )  =  ( F ` 
0 ) )
53 1elunit 11416 . . . 4  |-  1  e.  ( 0 [,] 1
)
54 simpr 461 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  y  =  s )
5554breq1d 4314 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( y  <_ 
( 1  /  2
)  <->  s  <_  (
1  /  2 ) ) )
56 simpl 457 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  x  =  1 )
5754oveq2d 6119 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( 2  x.  y )  =  ( 2  x.  s ) )
5856, 57oveq12d 6121 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x K ( 2  x.  y
) )  =  ( 1 K ( 2  x.  s ) ) )
5957oveq1d 6118 . . . . . . 7  |-  ( ( x  =  1  /\  y  =  s )  ->  ( ( 2  x.  y )  - 
1 )  =  ( ( 2  x.  s
)  -  1 ) )
6056, 59oveq12d 6121 . . . . . 6  |-  ( ( x  =  1  /\  y  =  s )  ->  ( x L ( ( 2  x.  y )  -  1 ) )  =  ( 1 L ( ( 2  x.  s )  -  1 ) ) )
6155, 58, 60ifbieq12d 3828 . . . . 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 6128 . . . . . 6  |-  ( 1 K ( 2  x.  s ) )  e. 
_V
63 ovex 6128 . . . . . 6  |-  ( 1 L ( ( 2  x.  s )  - 
1 ) )  e. 
_V
6462, 63ifex 3870 . . . . 5  |-  if ( s  <_  ( 1  /  2 ) ,  ( 1 K ( 2  x.  s ) ) ,  ( 1 L ( ( 2  x.  s )  - 
1 ) ) )  e.  _V
6561, 3, 64ovmpt2a 6233 . . . 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 663 . . 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 2449 . . . 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 2449 . . . 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 463 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  s  <_  ( 1  /  2
) )  ->  (
1 K ( 2  x.  s ) )  =  ( F ` 
1 ) )
7045simprd 463 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( G `
 1 ) )
7148simprd 463 . . . . 5  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( F `  1
)  =  ( G `
 1 ) )
7270, 71eqtr4d 2478 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 0 [,] 1
) )  /\  -.  s  <_  ( 1  / 
2 ) )  -> 
( 1 L ( ( 2  x.  s
)  -  1 ) )  =  ( F `
 1 ) )
7367, 68, 69, 72ifbothda 3836 . . 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 2475 . 2  |-  ( (
ph  /\  s  e.  ( 0 [,] 1
) )  ->  (
1 M s )  =  ( F ` 
1 ) )
751, 2, 13, 52, 74isphtpyd 20570 1  |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3803   class class class wbr 4304   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   0cc0 9294   1c1 9295    x. cmul 9299    <_ cle 9431    - cmin 9607    / cdiv 10005   2c2 10383   [,]cicc 11315  TopOnctopon 18511    Cn ccn 18840   IIcii 20463   Htpy chtpy 20551   PHtpycphtpy 20552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-mulf 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-fi 7673  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-ioo 11316  df-icc 11319  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-rest 14373  df-topn 14374  df-0g 14392  df-gsum 14393  df-topgen 14394  df-pt 14395  df-prds 14398  df-xrs 14452  df-qtop 14457  df-imas 14458  df-xps 14460  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-submnd 15477  df-mulg 15560  df-cntz 15847  df-cmn 16291  df-psmet 17821  df-xmet 17822  df-met 17823  df-bl 17824  df-mopn 17825  df-cnfld 17831  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-cld 18635  df-cn 18843  df-cnp 18844  df-tx 19147  df-hmeo 19340  df-xms 19907  df-ms 19908  df-tms 19909  df-ii 20465  df-htpy 20554  df-phtpy 20555
This theorem is referenced by:  phtpcer  20579
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