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Theorem chscllem4 23147
Description: Lemma for chscl 23148. (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 22745 . . . . 5  |-  ~~>v  : dom  ~~>v  --> ~H
2 ffun 5596 . . . . 5  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
31, 2ax-mp 5 . . . 4  |-  Fun  ~~>v
4 chscl.5 . . . 4  |-  ( ph  ->  H  ~~>v  u )
5 funbrfv 5768 . . . 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 5782 . . . . . 6  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( H `
 k ) ) )
97ffvelrnda 5873 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  e.  ( A  +H  B
) )
10 chscl.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CH )
11 chsh 22732 . . . . . . . . . . . 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 22732 . . . . . . . . . . . 12  |-  ( B  e.  CH  ->  B  e.  SH )
1513, 14syl 16 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  SH )
16 shsel 22821 . . . . . . . . . . 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 23146 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  =  ( F `  k ) )
33 chsscon2 23009 . . . . . . . . . . . . . . . 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 22826 . . . . . . . . . . . . . . . . 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 5581 . . . . . . . . . . . . . . . 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 22717 . . . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  ~H )
48 shss 22717 . . . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  ~H )
52 ax-hvcom 22509 . . . . . . . . . . . . . . 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 2470 . . . . . . . . . . . . 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 23146 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  =  ( G `  k ) )
5632, 55oveq12d 6102 . . . . . . . . . . 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 2470 . . . . . . . . . 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 2838 . . . . . . . 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 4298 . . . . . 6  |-  ( ph  ->  ( k  e.  NN  |->  ( H `  k ) )  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
628, 61eqtrd 2470 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
6310, 13, 24, 7, 4, 28chscllem1 23144 . . . . . . 7  |-  ( ph  ->  F : NN --> A )
64 fss 5602 . . . . . . 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 23144 . . . . . . 7  |-  ( ph  ->  G : NN --> B )
67 fss 5602 . . . . . . 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 23145 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ~~>v  )
70 funfvbrb 5846 . . . . . . . 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 23145 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ~~>v  )
74 funfvbrb 5846 . . . . . . . 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 2438 . . . . . 6  |-  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) )  =  ( k  e.  NN  |->  ( ( F `
 k )  +h  ( G `  k
) ) )
7865, 68, 72, 76, 77hlimadd 22700 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k )
) )  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
7962, 78eqbrtrd 4235 . . . 4  |-  ( ph  ->  H  ~~>v  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
80 funbrfv 5768 . . . 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 2472 . 2  |-  ( ph  ->  u  =  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
83 fvex 5745 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8483chlimi 22742 . . . 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 5745 . . . . 5  |-  (  ~~>v  `  G )  e.  _V
8786chlimi 22742 . . . 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 22827 . . . 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 2512 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 1726   E.wrex 2708    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   dom cdm 4881   Fun wfun 5451   -->wf 5453   ` cfv 5457  (class class class)co 6084   NNcn 10005   ~Hchil 22427    +h cva 22428    ~~>v chli 22435   SHcsh 22436   CHcch 22437   _|_cort 22438    +H cph 22439   proj 
hcpjh 22445
This theorem is referenced by:  chscl  23148
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 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592  ax-hcompl 22709
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 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-icc 10928  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cn 17296  df-cnp 17297  df-lm 17298  df-haus 17384  df-tx 17599  df-hmeo 17792  df-xms 18355  df-tms 18357  df-cau 19214  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-hnorm 22476  df-hba 22477  df-hvsub 22479  df-hlim 22480  df-hcau 22481  df-sh 22714  df-ch 22729  df-oc 22759  df-ch0 22760  df-shs 22815  df-pjh 22902
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