HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chscllem4 Structured version   Unicode version

Theorem chscllem4 23134
Description: Lemma for chscl 23135. (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 22732 . . . . 5  |-  ~~>v  : dom  ~~>v  --> ~H
2 ffun 5585 . . . . 5  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
31, 2ax-mp 8 . . . 4  |-  Fun  ~~>v
4 chscl.5 . . . 4  |-  ( ph  ->  H  ~~>v  u )
5 funbrfv 5757 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  u  -> 
(  ~~>v  `  H )  =  u ) )
63, 4, 5mpsyl 61 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  u )
7 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
87feqmptd 5771 . . . . . 6  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( H `
 k ) ) )
97ffvelrnda 5862 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  e.  ( A  +H  B
) )
10 chscl.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CH )
11 chsh 22719 . . . . . . . . . . . 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 22719 . . . . . . . . . . . 12  |-  ( B  e.  CH  ->  B  e.  SH )
1513, 14syl 16 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  SH )
16 shsel 22808 . . . . . . . . . . 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 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( H `  k )  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) ) )
1817biimpa 471 . . . . . . . . 9  |-  ( (
ph  /\  ( H `  k )  e.  ( A  +H  B ) )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
199, 18syldan 457 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
20 simp3 959 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( x  +h  y ) )
21 simp1l 981 . . . . . . . . . . . . . 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 982 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  k  e.  NN )
30 simp2l 983 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  A
)
31 simp2r 984 . . . . . . . . . . . . 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 23133 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  =  ( F `  k ) )
33 chsscon2 22996 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3413, 10, 33syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3524, 34mpbid 202 . . . . . . . . . . . . . 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 22813 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
3812, 15, 37syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
39 feq3 5570 . . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 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 22704 . . . . . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  ~H )
48 shss 22704 . . . . . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  ~H )
52 ax-hvcom 22496 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
5347, 51, 52syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( y  +h  x ) )
5420, 53eqtrd 2467 . . . . . . . . . . . . 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 23133 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  =  ( G `  k ) )
5632, 55oveq12d 6091 . . . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
58573exp 1152 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( H `  k )  =  ( x  +h  y )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) ) ) )
5958rexlimdvv 2828 . . . . . . . 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 4287 . . . . . 6  |-  ( ph  ->  ( k  e.  NN  |->  ( H `  k ) )  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
628, 61eqtrd 2467 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
6310, 13, 24, 7, 4, 28chscllem1 23131 . . . . . . 7  |-  ( ph  ->  F : NN --> A )
64 fss 5591 . . . . . . 7  |-  ( ( F : NN --> A  /\  A  C_  ~H )  ->  F : NN --> ~H )
6563, 45, 64syl2anc 643 . . . . . 6  |-  ( ph  ->  F : NN --> ~H )
6613, 10, 35, 41, 4, 43chscllem1 23131 . . . . . . 7  |-  ( ph  ->  G : NN --> B )
67 fss 5591 . . . . . . 7  |-  ( ( G : NN --> B  /\  B  C_  ~H )  ->  G : NN --> ~H )
6866, 49, 67syl2anc 643 . . . . . 6  |-  ( ph  ->  G : NN --> ~H )
6910, 13, 24, 7, 4, 28chscllem2 23132 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ~~>v  )
70 funfvbrb 5835 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
713, 70ax-mp 8 . . . . . . 7  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
7269, 71sylib 189 . . . . . 6  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
7313, 10, 35, 41, 4, 43chscllem2 23132 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ~~>v  )
74 funfvbrb 5835 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G )
) )
753, 74ax-mp 8 . . . . . . 7  |-  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G ) )
7673, 75sylib 189 . . . . . 6  |-  ( ph  ->  G  ~~>v  (  ~~>v  `  G
) )
77 eqid 2435 . . . . . 6  |-  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) )  =  ( k  e.  NN  |->  ( ( F `
 k )  +h  ( G `  k
) ) )
7865, 68, 72, 76, 77hlimadd 22687 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k )
) )  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
7962, 78eqbrtrd 4224 . . . 4  |-  ( ph  ->  H  ~~>v  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
80 funbrfv 5757 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
)  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v  `  G
) ) ) )
813, 79, 80mpsyl 61 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
826, 81eqtr3d 2469 . 2  |-  ( ph  ->  u  =  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
83 fvex 5734 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8483chlimi 22729 . . . 4  |-  ( ( A  e.  CH  /\  F : NN --> A  /\  F  ~~>v  (  ~~>v  `  F
) )  ->  (  ~~>v 
`  F )  e.  A )
8510, 63, 72, 84syl3anc 1184 . . 3  |-  ( ph  ->  (  ~~>v  `  F )  e.  A )
86 fvex 5734 . . . . 5  |-  (  ~~>v  `  G )  e.  _V
8786chlimi 22729 . . . 4  |-  ( ( B  e.  CH  /\  G : NN --> B  /\  G  ~~>v  (  ~~>v  `  G
) )  ->  (  ~~>v 
`  G )  e.  B )
8813, 66, 76, 87syl3anc 1184 . . 3  |-  ( ph  ->  (  ~~>v  `  G )  e.  B )
89 shsva 22814 . . . 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 643 . . 3  |-  ( ph  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
9185, 88, 90mp2and 661 . 2  |-  ( ph  ->  ( (  ~~>v  `  F
)  +h  (  ~~>v  `  G ) )  e.  ( A  +H  B
) )
9282, 91eqeltrd 2509 1  |-  ( ph  ->  u  e.  ( A  +H  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   dom cdm 4870   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   NNcn 9992   ~Hchil 22414    +h cva 22415    ~~>v chli 22422   SHcsh 22423   CHcch 22424   _|_cort 22425    +H cph 22426   proj 
hcpjh 22432
This theorem is referenced by:  chscl  23135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062  ax-hilex 22494  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvmulass 22502  ax-hvdistr1 22503  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579  ax-hcompl 22696
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cn 17283  df-cnp 17284  df-lm 17285  df-haus 17371  df-tx 17586  df-hmeo 17779  df-xms 18342  df-tms 18344  df-cau 19201  df-grpo 21771  df-gid 21772  df-ginv 21773  df-gdiv 21774  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-vs 22070  df-nmcv 22071  df-ims 22072  df-hnorm 22463  df-hba 22464  df-hvsub 22466  df-hlim 22467  df-hcau 22468  df-sh 22701  df-ch 22716  df-oc 22746  df-ch0 22747  df-shs 22802  df-pjh 22889
  Copyright terms: Public domain W3C validator