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Theorem chscllem4 22219
Description: Lemma for chscl 22220. (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 21817 . . . . 5  |-  ~~>v  : dom  ~~>v  --> ~H
2 ffun 5391 . . . . 5  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
31, 2ax-mp 8 . . . 4  |-  Fun  ~~>v
4 chscl.5 . . . 4  |-  ( ph  ->  H  ~~>v  u )
5 funbrfv 5561 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  u  -> 
(  ~~>v  `  H )  =  u ) )
63, 4, 5mpsyl 59 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  u )
7 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
87feqmptd 5575 . . . . . 6  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( H `
 k ) ) )
9 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( H : NN --> ( A  +H  B )  /\  k  e.  NN )  ->  ( H `  k
)  e.  ( A  +H  B ) )
107, 9sylan 457 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  e.  ( A  +H  B
) )
11 chscl.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CH )
12 chsh 21804 . . . . . . . . . . . 12  |-  ( A  e.  CH  ->  A  e.  SH )
1311, 12syl 15 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  SH )
14 chscl.2 . . . . . . . . . . . 12  |-  ( ph  ->  B  e.  CH )
15 chsh 21804 . . . . . . . . . . . 12  |-  ( B  e.  CH  ->  B  e.  SH )
1614, 15syl 15 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  SH )
17 shsel 21893 . . . . . . . . . . 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 ) ) )
1813, 16, 17syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( H `  k )  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) ) )
1918biimpa 470 . . . . . . . . 9  |-  ( (
ph  /\  ( H `  k )  e.  ( A  +H  B ) )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
2010, 19syldan 456 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
21 simp3 957 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( x  +h  y ) )
22 simp1l 979 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ph )
2322, 11syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  e.  CH )
2422, 14syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  e.  CH )
25 chscl.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  C_  ( _|_ `  A ) )
2622, 25syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  C_  ( _|_ `  A ) )
2722, 7syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H : NN --> ( A  +H  B
) )
2822, 4syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H  ~~>v  u )
29 chscl.6 . . . . . . . . . . . . 13  |-  F  =  ( n  e.  NN  |->  ( ( proj  h `  A ) `  ( H `  n )
) )
30 simp1r 980 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  k  e.  NN )
31 simp2l 981 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  A
)
32 simp2r 982 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  B
)
3323, 24, 26, 27, 28, 29, 30, 31, 32, 21chscllem3 22218 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  =  ( F `  k ) )
34 chsscon2 22081 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3514, 11, 34syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3625, 35mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  C_  ( _|_ `  B ) )
3722, 36syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  C_  ( _|_ `  B ) )
38 shscom 21898 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
3913, 16, 38syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
40 feq3 5377 . . . . . . . . . . . . . . . 16  |-  ( ( A  +H  B )  =  ( B  +H  A )  ->  ( H : NN --> ( A  +H  B )  <->  H : NN
--> ( B  +H  A
) ) )
4139, 40syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( H : NN --> ( A  +H  B
)  <->  H : NN --> ( B  +H  A ) ) )
427, 41mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  H : NN --> ( B  +H  A ) )
4322, 42syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H : NN --> ( B  +H  A
) )
44 chscl.7 . . . . . . . . . . . . 13  |-  G  =  ( n  e.  NN  |->  ( ( proj  h `  B ) `  ( H `  n )
) )
45 shss 21789 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  SH  ->  A  C_ 
~H )
4613, 45syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  C_  ~H )
4722, 46syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  C_  ~H )
4847, 31sseldd 3181 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  ~H )
49 shss 21789 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  SH  ->  B  C_ 
~H )
5016, 49syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  C_  ~H )
5122, 50syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  C_  ~H )
5251, 32sseldd 3181 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  ~H )
53 ax-hvcom 21581 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
5448, 52, 53syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( y  +h  x ) )
5521, 54eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( y  +h  x ) )
5624, 23, 37, 43, 28, 44, 30, 32, 31, 55chscllem3 22218 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  =  ( G `  k ) )
5733, 56oveq12d 5876 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
5821, 57eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
59583exp 1150 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( H `  k )  =  ( x  +h  y )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) ) ) )
6059rexlimdvv 2673 . . . . . . . 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
) ) ) )
6120, 60mpd 14 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( ( F `  k )  +h  ( G `  k )
) )
6261mpteq2dva 4106 . . . . . 6  |-  ( ph  ->  ( k  e.  NN  |->  ( H `  k ) )  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
638, 62eqtrd 2315 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
6411, 14, 25, 7, 4, 29chscllem1 22216 . . . . . . 7  |-  ( ph  ->  F : NN --> A )
65 fss 5397 . . . . . . 7  |-  ( ( F : NN --> A  /\  A  C_  ~H )  ->  F : NN --> ~H )
6664, 46, 65syl2anc 642 . . . . . 6  |-  ( ph  ->  F : NN --> ~H )
6714, 11, 36, 42, 4, 44chscllem1 22216 . . . . . . 7  |-  ( ph  ->  G : NN --> B )
68 fss 5397 . . . . . . 7  |-  ( ( G : NN --> B  /\  B  C_  ~H )  ->  G : NN --> ~H )
6967, 50, 68syl2anc 642 . . . . . 6  |-  ( ph  ->  G : NN --> ~H )
7011, 14, 25, 7, 4, 29chscllem2 22217 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ~~>v  )
71 funfvbrb 5638 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
723, 71ax-mp 8 . . . . . . 7  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
7370, 72sylib 188 . . . . . 6  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
7414, 11, 36, 42, 4, 44chscllem2 22217 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ~~>v  )
75 funfvbrb 5638 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G )
) )
763, 75ax-mp 8 . . . . . . 7  |-  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G ) )
7774, 76sylib 188 . . . . . 6  |-  ( ph  ->  G  ~~>v  (  ~~>v  `  G
) )
78 eqid 2283 . . . . . 6  |-  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) )  =  ( k  e.  NN  |->  ( ( F `
 k )  +h  ( G `  k
) ) )
7966, 69, 73, 77, 78hlimadd 21772 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k )
) )  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
8063, 79eqbrtrd 4043 . . . 4  |-  ( ph  ->  H  ~~>v  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
81 funbrfv 5561 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
)  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v  `  G
) ) ) )
823, 80, 81mpsyl 59 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
836, 82eqtr3d 2317 . 2  |-  ( ph  ->  u  =  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
84 fvex 5539 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8584chlimi 21814 . . . 4  |-  ( ( A  e.  CH  /\  F : NN --> A  /\  F  ~~>v  (  ~~>v  `  F
) )  ->  (  ~~>v 
`  F )  e.  A )
8611, 64, 73, 85syl3anc 1182 . . 3  |-  ( ph  ->  (  ~~>v  `  F )  e.  A )
87 fvex 5539 . . . . 5  |-  (  ~~>v  `  G )  e.  _V
8887chlimi 21814 . . . 4  |-  ( ( B  e.  CH  /\  G : NN --> B  /\  G  ~~>v  (  ~~>v  `  G
) )  ->  (  ~~>v 
`  G )  e.  B )
8914, 67, 77, 88syl3anc 1182 . . 3  |-  ( ph  ->  (  ~~>v  `  G )  e.  B )
90 shsva 21899 . . . 4  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
9113, 16, 90syl2anc 642 . . 3  |-  ( ph  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
9286, 89, 91mp2and 660 . 2  |-  ( ph  ->  ( (  ~~>v  `  F
)  +h  (  ~~>v  `  G ) )  e.  ( A  +H  B
) )
9383, 92eqeltrd 2357 1  |-  ( ph  ->  u  e.  ( A  +H  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   NNcn 9746   ~Hchil 21499    +h cva 21500    ~~>v chli 21507   SHcsh 21508   CHcch 21509   _|_cort 21510    +H cph 21511   proj 
hcpjh 21517
This theorem is referenced by:  chscl  22220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-xms 17885  df-tms 17887  df-cau 18682  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-pjh 21974
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