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Theorem chscllem4 26536
Description: Lemma for chscl 26537. (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 26133 . . . . 5  |-  ~~>v  : dom  ~~>v  --> ~H
2 ffun 5723 . . . . 5  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
31, 2ax-mp 5 . . . 4  |-  Fun  ~~>v
4 chscl.5 . . . 4  |-  ( ph  ->  H  ~~>v  u )
5 funbrfv 5896 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  u  -> 
(  ~~>v  `  H )  =  u ) )
63, 4, 5mpsyl 63 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  u )
7 chscl.4 . . . . . . 7  |-  ( ph  ->  H : NN --> ( A  +H  B ) )
87feqmptd 5911 . . . . . 6  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( H `
 k ) ) )
97ffvelrnda 6016 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  e.  ( A  +H  B
) )
10 chscl.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  CH )
11 chsh 26120 . . . . . . . . . . . 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 26120 . . . . . . . . . . . 12  |-  ( B  e.  CH  ->  B  e.  SH )
1513, 14syl 16 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  SH )
16 shsel 26210 . . . . . . . . . . 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 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( H `  k )  e.  ( A  +H  B )  <->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) ) )
1817biimpa 484 . . . . . . . . 9  |-  ( (
ph  /\  ( H `  k )  e.  ( A  +H  B ) )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
199, 18syldan 470 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  E. x  e.  A  E. y  e.  B  ( H `  k )  =  ( x  +h  y ) )
20 simp3 999 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( x  +h  y ) )
21 simp1l 1021 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  k  e.  NN )
30 simp2l 1023 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  A
)
31 simp2r 1024 . . . . . . . . . . . . 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 26535 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  =  ( F `  k ) )
33 chsscon2 26398 . . . . . . . . . . . . . . . 16  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3413, 10, 33syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( B  C_  ( _|_ `  A )  <->  A  C_  ( _|_ `  B ) ) )
3524, 34mpbid 210 . . . . . . . . . . . . . 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 26215 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( A  +H  B
)  =  ( B  +H  A ) )
3812, 15, 37syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( A  +H  B
)  =  ( B  +H  A ) )
3938feq3d 5709 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( H : NN --> ( A  +H  B
)  <->  H : NN --> ( B  +H  A ) ) )
407, 39mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  H : NN --> ( B  +H  A ) )
4121, 40syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  H : NN --> ( B  +H  A
) )
42 chscl.7 . . . . . . . . . . . . 13  |-  G  =  ( n  e.  NN  |->  ( ( proj h `  B ) `  ( H `  n )
) )
43 shss 26105 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  SH  ->  A  C_ 
~H )
4412, 43syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A  C_  ~H )
4521, 44syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  A  C_  ~H )
4645, 30sseldd 3490 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  x  e.  ~H )
47 shss 26105 . . . . . . . . . . . . . . . . . 18  |-  ( B  e.  SH  ->  B  C_ 
~H )
4815, 47syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  B  C_  ~H )
4921, 48syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  B  C_  ~H )
5049, 31sseldd 3490 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  e.  ~H )
51 ax-hvcom 25896 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  =  ( y  +h  x ) )
5246, 50, 51syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( y  +h  x ) )
5320, 52eqtrd 2484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( y  +h  x ) )
5423, 22, 36, 41, 27, 42, 29, 31, 30, 53chscllem3 26535 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  y  =  ( G `  k ) )
5532, 54oveq12d 6299 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( x  +h  y )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
5620, 55eqtrd 2484 . . . . . . . . . 10  |-  ( ( ( ph  /\  k  e.  NN )  /\  (
x  e.  A  /\  y  e.  B )  /\  ( H `  k
)  =  ( x  +h  y ) )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) )
57563exp 1196 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( H `  k )  =  ( x  +h  y )  ->  ( H `  k )  =  ( ( F `  k
)  +h  ( G `
 k ) ) ) ) )
5857rexlimdvv 2941 . . . . . . . 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
) ) ) )
5919, 58mpd 15 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  ( H `
 k )  =  ( ( F `  k )  +h  ( G `  k )
) )
6059mpteq2dva 4523 . . . . . 6  |-  ( ph  ->  ( k  e.  NN  |->  ( H `  k ) )  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
618, 60eqtrd 2484 . . . . 5  |-  ( ph  ->  H  =  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) ) )
6210, 13, 24, 7, 4, 28chscllem1 26533 . . . . . . 7  |-  ( ph  ->  F : NN --> A )
6362, 44fssd 5730 . . . . . 6  |-  ( ph  ->  F : NN --> ~H )
6413, 10, 35, 40, 4, 42chscllem1 26533 . . . . . . 7  |-  ( ph  ->  G : NN --> B )
6564, 48fssd 5730 . . . . . 6  |-  ( ph  ->  G : NN --> ~H )
6610, 13, 24, 7, 4, 28chscllem2 26534 . . . . . . 7  |-  ( ph  ->  F  e.  dom  ~~>v  )
67 funfvbrb 5985 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F )
) )
683, 67ax-mp 5 . . . . . . 7  |-  ( F  e.  dom  ~~>v  <->  F  ~~>v  (  ~~>v  `  F ) )
6966, 68sylib 196 . . . . . 6  |-  ( ph  ->  F  ~~>v  (  ~~>v  `  F
) )
7013, 10, 35, 40, 4, 42chscllem2 26534 . . . . . . 7  |-  ( ph  ->  G  e.  dom  ~~>v  )
71 funfvbrb 5985 . . . . . . . 8  |-  ( Fun  ~~>v 
->  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G )
) )
723, 71ax-mp 5 . . . . . . 7  |-  ( G  e.  dom  ~~>v  <->  G  ~~>v  (  ~~>v  `  G ) )
7370, 72sylib 196 . . . . . 6  |-  ( ph  ->  G  ~~>v  (  ~~>v  `  G
) )
74 eqid 2443 . . . . . 6  |-  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k ) ) )  =  ( k  e.  NN  |->  ( ( F `
 k )  +h  ( G `  k
) ) )
7563, 65, 69, 73, 74hlimadd 26088 . . . . 5  |-  ( ph  ->  ( k  e.  NN  |->  ( ( F `  k )  +h  ( G `  k )
) )  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
7661, 75eqbrtrd 4457 . . . 4  |-  ( ph  ->  H  ~~>v  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
77 funbrfv 5896 . . . 4  |-  ( Fun  ~~>v 
->  ( H  ~~>v  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
)  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v  `  G
) ) ) )
783, 76, 77mpsyl 63 . . 3  |-  ( ph  ->  (  ~~>v  `  H )  =  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) ) )
796, 78eqtr3d 2486 . 2  |-  ( ph  ->  u  =  ( ( 
~~>v  `  F )  +h  (  ~~>v  `  G )
) )
80 fvex 5866 . . . . 5  |-  (  ~~>v  `  F )  e.  _V
8180chlimi 26130 . . . 4  |-  ( ( A  e.  CH  /\  F : NN --> A  /\  F  ~~>v  (  ~~>v  `  F
) )  ->  (  ~~>v 
`  F )  e.  A )
8210, 62, 69, 81syl3anc 1229 . . 3  |-  ( ph  ->  (  ~~>v  `  F )  e.  A )
83 fvex 5866 . . . . 5  |-  (  ~~>v  `  G )  e.  _V
8483chlimi 26130 . . . 4  |-  ( ( B  e.  CH  /\  G : NN --> B  /\  G  ~~>v  (  ~~>v  `  G
) )  ->  (  ~~>v 
`  G )  e.  B )
8513, 64, 73, 84syl3anc 1229 . . 3  |-  ( ph  ->  (  ~~>v  `  G )  e.  B )
86 shsva 26216 . . . 4  |-  ( ( A  e.  SH  /\  B  e.  SH )  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
8712, 15, 86syl2anc 661 . . 3  |-  ( ph  ->  ( ( (  ~~>v  `  F )  e.  A  /\  (  ~~>v  `  G
)  e.  B )  ->  ( (  ~~>v  `  F )  +h  (  ~~>v 
`  G ) )  e.  ( A  +H  B ) ) )
8882, 85, 87mp2and 679 . 2  |-  ( ph  ->  ( (  ~~>v  `  F
)  +h  (  ~~>v  `  G ) )  e.  ( A  +H  B
) )
8979, 88eqeltrd 2531 1  |-  ( ph  ->  u  e.  ( A  +H  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   E.wrex 2794    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   NNcn 10543   ~Hchil 25814    +h cva 25815    ~~>v chli 25822   SHcsh 25823   CHcch 25824   _|_cort 25825    +H cph 25826   proj hcpjh 25832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575  ax-hilex 25894  ax-hfvadd 25895  ax-hvcom 25896  ax-hvass 25897  ax-hv0cl 25898  ax-hvaddid 25899  ax-hfvmul 25900  ax-hvmulid 25901  ax-hvmulass 25902  ax-hvdistr1 25903  ax-hvdistr2 25904  ax-hvmul0 25905  ax-hfi 25974  ax-his1 25977  ax-his2 25978  ax-his3 25979  ax-his4 25980  ax-hcompl 26097
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-icc 11547  df-fz 11684  df-fzo 11807  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cn 19706  df-cnp 19707  df-lm 19708  df-haus 19794  df-tx 20041  df-hmeo 20234  df-xms 20801  df-tms 20803  df-cau 21673  df-grpo 25171  df-gid 25172  df-ginv 25173  df-gdiv 25174  df-ablo 25262  df-vc 25417  df-nv 25463  df-va 25466  df-ba 25467  df-sm 25468  df-0v 25469  df-vs 25470  df-nmcv 25471  df-ims 25472  df-hnorm 25863  df-hba 25864  df-hvsub 25866  df-hlim 25867  df-hcau 25868  df-sh 26102  df-ch 26117  df-oc 26148  df-ch0 26149  df-shs 26204  df-pjh 26291
This theorem is referenced by:  chscl  26537
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