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Theorem chscllem4 23173
Description: Lemma for chscl 23174. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
chscl.1  |-  ( ph  ->  A  e.  CH )
chscl.2  |-  ( ph  ->  B  e.  CH )
chscl.3  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
chscl.4  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
chscl.5  |-  ( ph  ->  H  ~~>v  u )
chscl.6  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
chscl.7  |-  G  =  ( n  e.  NN  |->  ( ( proj  h `  B ) `  ( H `  n )
) )
Assertion
Ref Expression
chscllem4  |-  ( ph  ->  u  e.  ( A  +H  B ) )
Distinct variable groups:    u, n, A    ph, n    B, n, u    n, H, u
Allowed substitution hints:    ph( u)    F( u, n)    G( u, n)

Proof of Theorem chscllem4
Dummy variables  x  y  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlimf 22771 . . . . 5  |-  ~~>v  : dom  ~~>v  --> ~H
2 ffun 5622 . . . . 5  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
31, 2ax-mp 5 . . . 4  |-  Fun  ~~>v
4 chscl.5 . . . 4  |-  ( ph  ->  H  ~~>v  u )
5 funbrfv 5794 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  u  -> 
(  ~~>v  `  H )  =  u ) )
63, 4, 5mpsyl 62 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  u )
7 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
87feqmptd 5808 . . . . . 6  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( H `
 k ) ) )
97ffvelrnda 5899 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  e.  ( A  +H  B
) )
10 chscl.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CH )
11 chsh 22758 . . . . . . . . . . . 12  |-  ( A  e.  CH  ->  A  e.  SH )
1210, 11syl 16 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  SH )
13 chscl.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CH )
14 chsh 22758 . . . . . . . . . . . 12  |-  ( B  e.  CH  ->  B  e.  SH )
1513, 14syl 16 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  SH )
16 shsel 22847 . . . . . . . . . . 11  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( ( H `  k )  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) ) )
1712, 15, 16syl2anc 644 . . . . . . . . . 10  |-  ( ph  ->  ( ( H `  k )  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) ) )
1817biimpa 472 . . . . . . . . 9  |-  ( (
ph  /\  ( H `  k )  e.  ( A  +H  B ) )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
199, 18syldan 458 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
20 simp3 960 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( x  +h  y ) )
21 simp1l 982 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ph )
2221, 10syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  e.  CH )
2321, 13syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  e.  CH )
24 chscl.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
2521, 24syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  C_  ( _|_ `  A ) )
2621, 7syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H : NN --> ( A  +H  B
) )
2721, 4syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H  ~~>v  u )
28 chscl.6 . . . . . . . . . . . . 13  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
29 simp1r 983 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  k  e.  NN )
30 simp2l 984 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  A
)
31 simp2r 985 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  B
)
3222, 23, 25, 26, 27, 28, 29, 30, 31, 20chscllem3 23172 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  =  ( F `  k ) )
33 chsscon2 23035 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3413, 10, 33syl2anc 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3524, 34mpbid 203 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  C_  ( _|_ `  B ) )
3621, 35syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  C_  ( _|_ `  B ) )
37 shscom 22852 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
3812, 15, 37syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
39 feq3 5607 . . . . . . . . . . . . . . . 16  |-  ( ( A  +H  B )  =  ( B  +H  A )  ->  ( H : NN --> ( A  +H  B )  <->  H : NN
--> ( B  +H  A
) ) )
4038, 39syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( H : NN --> ( A  +H  B
)  <->  H : NN --> ( B  +H  A ) ) )
417, 40mpbid 203 . . . . . . . . . . . . . 14  |-  ( ph  ->  H : NN --> ( B  +H  A ) )
4221, 41syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H : NN --> ( B  +H  A
) )
43 chscl.7 . . . . . . . . . . . . 13  |-  G  =  ( n  e.  NN  |->  ( ( proj  h `  B ) `  ( H `  n )
) )
44 shss 22743 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  SH  ->  A  C_ 
~H )
4512, 44syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  C_  ~H )
4621, 45syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  C_  ~H )
4746, 30sseldd 3335 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  ~H )
48 shss 22743 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  SH  ->  B  C_ 
~H )
4915, 48syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  C_  ~H )
5021, 49syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  C_  ~H )
5150, 31sseldd 3335 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  ~H )
52 ax-hvcom 22535 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
5347, 51, 52syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( y  +h  x ) )
5420, 53eqtrd 2474 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( y  +h  x ) )
5523, 22, 36, 42, 27, 43, 29, 31, 30, 54chscllem3 23172 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  =  ( G `  k ) )
5632, 55oveq12d 6128 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
5720, 56eqtrd 2474 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
58573exp 1153 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( H `  k )  =  ( x  +h  y )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) ) ) )
5958rexlimdvv 2842 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y )  ->  ( H `  k )  =  ( ( F `
 k )  +h  ( G `  k
) ) ) )
6019, 59mpd 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( ( F `  k )  +h  ( G `  k )
) )
6160mpteq2dva 4320 . . . . . 6  |-  ( ph  ->  ( k  e.  NN  |->  ( H `  k ) )  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
628, 61eqtrd 2474 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
6310, 13, 24, 7, 4, 28chscllem1 23170 . . . . . . 7  |-  ( ph  ->  F : NN --> A )
64 fss 5628 . . . . . . 7  |-  ( ( F : NN --> A  /\  A  C_  ~H )  ->  F : NN --> ~H )
6563, 45, 64syl2anc 644 . . . . . 6  |-  ( ph  ->  F : NN --> ~H )
6613, 10, 35, 41, 4, 43chscllem1 23170 . . . . . . 7  |-  ( ph  ->  G : NN --> B )
67 fss 5628 . . . . . . 7  |-  ( ( G : NN --> B  /\  B  C_  ~H )  ->  G : NN --> ~H )
6866, 49, 67syl2anc 644 . . . . . 6  |-  ( ph  ->  G : NN --> ~H )
6910, 13, 24, 7, 4, 28chscllem2 23171 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ~~>v  )
70 funfvbrb 5872 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
713, 70ax-mp 5 . . . . . . 7  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
7269, 71sylib 190 . . . . . 6  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
7313, 10, 35, 41, 4, 43chscllem2 23171 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ~~>v  )
74 funfvbrb 5872 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G )
) )
753, 74ax-mp 5 . . . . . . 7  |-  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G ) )
7673, 75sylib 190 . . . . . 6  |-  ( ph  ->  G  ~~>v  (  ~~>v  `  G
) )
77 eqid 2442 . . . . . 6  |-  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) )  =  ( k  e.  NN  |->  ( ( F `
 k )  +h  ( G `  k
) ) )
7865, 68, 72, 76, 77hlimadd 22726 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k )
) )  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
7962, 78eqbrtrd 4257 . . . 4  |-  ( ph  ->  H  ~~>v  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
80 funbrfv 5794 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
)  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v  `  G
) ) ) )
813, 79, 80mpsyl 62 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
826, 81eqtr3d 2476 . 2  |-  ( ph  ->  u  =  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
83 fvex 5771 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8483chlimi 22768 . . . 4  |-  ( ( A  e.  CH  /\  F : NN --> A  /\  F  ~~>v  (  ~~>v  `  F
) )  ->  (  ~~>v 
`  F )  e.  A )
8510, 63, 72, 84syl3anc 1185 . . 3  |-  ( ph  ->  (  ~~>v  `  F )  e.  A )
86 fvex 5771 . . . . 5  |-  (  ~~>v  `  G )  e.  _V
8786chlimi 22768 . . . 4  |-  ( ( B  e.  CH  /\  G : NN --> B  /\  G  ~~>v  (  ~~>v  `  G
) )  ->  (  ~~>v 
`  G )  e.  B )
8813, 66, 76, 87syl3anc 1185 . . 3  |-  ( ph  ->  (  ~~>v  `  G )  e.  B )
89 shsva 22853 . . . 4  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
9012, 15, 89syl2anc 644 . . 3  |-  ( ph  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
9185, 88, 90mp2and 662 . 2  |-  ( ph  ->  ( (  ~~>v  `  F
)  +h  (  ~~>v  `  G ) )  e.  ( A  +H  B
) )
9282, 91eqeltrd 2516 1  |-  ( ph  ->  u  e.  ( A  +H  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   E.wrex 2712    C_ wss 3306   class class class wbr 4237    e. cmpt 4291   dom cdm 4907   Fun wfun 5477   -->wf 5479   ` cfv 5483  (class class class)co 6110   NNcn 10031   ~Hchil 22453    +h cva 22454    ~~>v chli 22461   SHcsh 22462   CHcch 22463   _|_cort 22464    +H cph 22465   proj 
hcpjh 22471
This theorem is referenced by:  chscl  23174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101  ax-hilex 22533  ax-hfvadd 22534  ax-hvcom 22535  ax-hvass 22536  ax-hv0cl 22537  ax-hvaddid 22538  ax-hfvmul 22539  ax-hvmulid 22540  ax-hvmulass 22541  ax-hvdistr1 22542  ax-hvdistr2 22543  ax-hvmul0 22544  ax-hfi 22612  ax-his1 22615  ax-his2 22616  ax-his3 22617  ax-his4 22618  ax-hcompl 22735
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-icc 10954  df-fz 11075  df-fzo 11167  df-seq 11355  df-exp 11414  df-hash 11650  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-mulg 14846  df-cntz 15147  df-cmn 15445  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cn 17322  df-cnp 17323  df-lm 17324  df-haus 17410  df-tx 17625  df-hmeo 17818  df-xms 18381  df-tms 18383  df-cau 19240  df-grpo 21810  df-gid 21811  df-ginv 21812  df-gdiv 21813  df-ablo 21901  df-vc 22056  df-nv 22102  df-va 22105  df-ba 22106  df-sm 22107  df-0v 22108  df-vs 22109  df-nmcv 22110  df-ims 22111  df-hnorm 22502  df-hba 22503  df-hvsub 22505  df-hlim 22506  df-hcau 22507  df-sh 22740  df-ch 22755  df-oc 22785  df-ch0 22786  df-shs 22841  df-pjh 22928
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