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Theorem chirredlem3 25747
Description: Lemma for chirredi 25749. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
chirred.1  |-  A  e. 
CH
chirred.2  |-  ( x  e.  CH  ->  A  C_H  x )
Assertion
Ref Expression
chirredlem3  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  C_  A  ->  r  =  p ) )
Distinct variable group:    q, p, r, x, A

Proof of Theorem chirredlem3
StepHypRef Expression
1 atelch 25699 . . 3  |-  ( q  e. HAtoms  ->  q  e.  CH )
2 chirred.1 . . . . . . . . . . . 12  |-  A  e. 
CH
32chirredlem2 25746 . . . . . . . . . . 11  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( ( _|_ `  r
)  i^i  ( p  vH  q ) )  =  q )
43oveq2d 6102 . . . . . . . . . 10  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( r  vH  (
( _|_ `  r
)  i^i  ( p  vH  q ) ) )  =  ( r  vH  q ) )
5 atelch 25699 . . . . . . . . . . . . . 14  |-  ( r  e. HAtoms  ->  r  e.  CH )
65adantr 465 . . . . . . . . . . . . 13  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  r  e.  CH )
7 atelch 25699 . . . . . . . . . . . . . . 15  |-  ( p  e. HAtoms  ->  p  e.  CH )
8 chjcl 24711 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( p  vH  q
)  e.  CH )
97, 8sylan 471 . . . . . . . . . . . . . 14  |-  ( ( p  e. HAtoms  /\  q  e.  CH )  ->  (
p  vH  q )  e.  CH )
109ad2ant2r 746 . . . . . . . . . . . . 13  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( p  vH  q
)  e.  CH )
11 id 22 . . . . . . . . . . . . 13  |-  ( r 
C_  ( p  vH  q )  ->  r  C_  ( p  vH  q
) )
12 pjoml2 24965 . . . . . . . . . . . . 13  |-  ( ( r  e.  CH  /\  ( p  vH  q
)  e.  CH  /\  r  C_  ( p  vH  q ) )  -> 
( r  vH  (
( _|_ `  r
)  i^i  ( p  vH  q ) ) )  =  ( p  vH  q ) )
136, 10, 11, 12syl3an 1260 . . . . . . . . . . . 12  |-  ( ( ( r  e. HAtoms  /\  r  C_  A )  /\  (
( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  /\  r  C_  ( p  vH  q ) )  -> 
( r  vH  (
( _|_ `  r
)  i^i  ( p  vH  q ) ) )  =  ( p  vH  q ) )
14133com12 1191 . . . . . . . . . . 11  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) )  ->  (
r  vH  ( ( _|_ `  r )  i^i  ( p  vH  q
) ) )  =  ( p  vH  q
) )
15143expb 1188 . . . . . . . . . 10  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( r  vH  (
( _|_ `  r
)  i^i  ( p  vH  q ) ) )  =  ( p  vH  q ) )
164, 15eqtr3d 2472 . . . . . . . . 9  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( r  vH  q
)  =  ( p  vH  q ) )
1716ineq2d 3547 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  (
r  vH  q )
)  =  ( A  i^i  ( p  vH  q ) ) )
18 breq2 4291 . . . . . . . . . . . . . . . 16  |-  ( x  =  r  ->  ( A  C_H  x  <->  A  C_H  r ) )
19 chirred.2 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  A  C_H  x )
2018, 19vtoclga 3031 . . . . . . . . . . . . . . 15  |-  ( r  e.  CH  ->  A  C_H  r )
21 breq2 4291 . . . . . . . . . . . . . . . 16  |-  ( x  =  q  ->  ( A  C_H  x  <->  A  C_H  q ) )
2221, 19vtoclga 3031 . . . . . . . . . . . . . . 15  |-  ( q  e.  CH  ->  A  C_H  q )
2320, 22anim12i 566 . . . . . . . . . . . . . 14  |-  ( ( r  e.  CH  /\  q  e.  CH )  ->  ( A  C_H  r  /\  A  C_H  q ) )
24 fh1 24972 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  CH  /\  r  e.  CH  /\  q  e.  CH )  /\  ( A  C_H  r  /\  A  C_H  q ) )  ->  ( A  i^i  ( r  vH  q
) )  =  ( ( A  i^i  r
)  vH  ( A  i^i  q ) ) )
252, 24mp3anl1 1308 . . . . . . . . . . . . . 14  |-  ( ( ( r  e.  CH  /\  q  e.  CH )  /\  ( A  C_H  r  /\  A  C_H  q ) )  ->  ( A  i^i  ( r  vH  q
) )  =  ( ( A  i^i  r
)  vH  ( A  i^i  q ) ) )
2623, 25mpdan 668 . . . . . . . . . . . . 13  |-  ( ( r  e.  CH  /\  q  e.  CH )  ->  ( A  i^i  (
r  vH  q )
)  =  ( ( A  i^i  r )  vH  ( A  i^i  q ) ) )
275, 26sylan 471 . . . . . . . . . . . 12  |-  ( ( r  e. HAtoms  /\  q  e.  CH )  ->  ( A  i^i  ( r  vH  q ) )  =  ( ( A  i^i  r )  vH  ( A  i^i  q ) ) )
2827ancoms 453 . . . . . . . . . . 11  |-  ( ( q  e.  CH  /\  r  e. HAtoms )  ->  ( A  i^i  ( r  vH  q ) )  =  ( ( A  i^i  r )  vH  ( A  i^i  q
) ) )
2928adantrr 716 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
( A  i^i  (
r  vH  q )
)  =  ( ( A  i^i  r )  vH  ( A  i^i  q ) ) )
3029ad2ant2r 746 . . . . . . . . 9  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  (
p  vH  q )
) )  ->  ( A  i^i  ( r  vH  q ) )  =  ( ( A  i^i  r )  vH  ( A  i^i  q ) ) )
3130adantll 713 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  (
r  vH  q )
)  =  ( ( A  i^i  r )  vH  ( A  i^i  q ) ) )
32 breq2 4291 . . . . . . . . . . . . . 14  |-  ( x  =  p  ->  ( A  C_H  x  <->  A  C_H  p ) )
3332, 19vtoclga 3031 . . . . . . . . . . . . 13  |-  ( p  e.  CH  ->  A  C_H  p )
3433, 22anim12i 566 . . . . . . . . . . . 12  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( A  C_H  p  /\  A  C_H  q ) )
35 fh1 24972 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  CH  /\  p  e.  CH  /\  q  e.  CH )  /\  ( A  C_H  p  /\  A  C_H  q ) )  ->  ( A  i^i  ( p  vH  q
) )  =  ( ( A  i^i  p
)  vH  ( A  i^i  q ) ) )
362, 35mp3anl1 1308 . . . . . . . . . . . 12  |-  ( ( ( p  e.  CH  /\  q  e.  CH )  /\  ( A  C_H  p  /\  A  C_H  q ) )  ->  ( A  i^i  ( p  vH  q
) )  =  ( ( A  i^i  p
)  vH  ( A  i^i  q ) ) )
3734, 36mpdan 668 . . . . . . . . . . 11  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( A  i^i  (
p  vH  q )
)  =  ( ( A  i^i  p )  vH  ( A  i^i  q ) ) )
387, 37sylan 471 . . . . . . . . . 10  |-  ( ( p  e. HAtoms  /\  q  e.  CH )  ->  ( A  i^i  ( p  vH  q ) )  =  ( ( A  i^i  p )  vH  ( A  i^i  q ) ) )
3938ad2ant2r 746 . . . . . . . . 9  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( A  i^i  (
p  vH  q )
)  =  ( ( A  i^i  p )  vH  ( A  i^i  q ) ) )
4039adantr 465 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  (
p  vH  q )
)  =  ( ( A  i^i  p )  vH  ( A  i^i  q ) ) )
4117, 31, 403eqtr3d 2478 . . . . . . 7  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( ( A  i^i  r )  vH  ( A  i^i  q ) )  =  ( ( A  i^i  p )  vH  ( A  i^i  q
) ) )
42 sseqin2 3564 . . . . . . . . . . 11  |-  ( r 
C_  A  <->  ( A  i^i  r )  =  r )
4342biimpi 194 . . . . . . . . . 10  |-  ( r 
C_  A  ->  ( A  i^i  r )  =  r )
4443ad2antlr 726 . . . . . . . . 9  |-  ( ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) )  ->  ( A  i^i  r )  =  r )
4544adantl 466 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  r
)  =  r )
46 incom 3538 . . . . . . . . . 10  |-  ( A  i^i  q )  =  ( q  i^i  A
)
47 chsh 24578 . . . . . . . . . . . 12  |-  ( q  e.  CH  ->  q  e.  SH )
482chshii 24581 . . . . . . . . . . . 12  |-  A  e.  SH
49 orthin 24800 . . . . . . . . . . . 12  |-  ( ( q  e.  SH  /\  A  e.  SH )  ->  ( q  C_  ( _|_ `  A )  -> 
( q  i^i  A
)  =  0H ) )
5047, 48, 49sylancl 662 . . . . . . . . . . 11  |-  ( q  e.  CH  ->  (
q  C_  ( _|_ `  A )  ->  (
q  i^i  A )  =  0H ) )
5150imp 429 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  q  C_  ( _|_ `  A
) )  ->  (
q  i^i  A )  =  0H )
5246, 51syl5eq 2482 . . . . . . . . 9  |-  ( ( q  e.  CH  /\  q  C_  ( _|_ `  A
) )  ->  ( A  i^i  q )  =  0H )
5352ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  q
)  =  0H )
5445, 53oveq12d 6104 . . . . . . 7  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( ( A  i^i  r )  vH  ( A  i^i  q ) )  =  ( r  vH  0H ) )
55 sseqin2 3564 . . . . . . . . . . 11  |-  ( p 
C_  A  <->  ( A  i^i  p )  =  p )
5655biimpi 194 . . . . . . . . . 10  |-  ( p 
C_  A  ->  ( A  i^i  p )  =  p )
5756adantl 466 . . . . . . . . 9  |-  ( ( p  e. HAtoms  /\  p  C_  A )  ->  ( A  i^i  p )  =  p )
5857ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( A  i^i  p
)  =  p )
5958, 53oveq12d 6104 . . . . . . 7  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( ( A  i^i  p )  vH  ( A  i^i  q ) )  =  ( p  vH  0H ) )
6041, 54, 593eqtr3d 2478 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( r  vH  0H )  =  ( p  vH  0H ) )
61 chj0 24851 . . . . . . . . 9  |-  ( r  e.  CH  ->  (
r  vH  0H )  =  r )
625, 61syl 16 . . . . . . . 8  |-  ( r  e. HAtoms  ->  ( r  vH  0H )  =  r
)
6362ad2antrr 725 . . . . . . 7  |-  ( ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) )  ->  (
r  vH  0H )  =  r )
6463adantl 466 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( r  vH  0H )  =  r )
65 chj0 24851 . . . . . . . 8  |-  ( p  e.  CH  ->  (
p  vH  0H )  =  p )
667, 65syl 16 . . . . . . 7  |-  ( p  e. HAtoms  ->  ( p  vH  0H )  =  p
)
6766ad3antrrr 729 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( p  vH  0H )  =  p )
6860, 64, 673eqtr3d 2478 . . . . 5  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
r  =  p )
6968exp44 613 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( r  e. HAtoms  ->  ( r  C_  A  ->  ( r  C_  ( p  vH  q )  ->  r  =  p ) ) ) )
7069com34 83 . . 3  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( r  e. HAtoms  ->  ( r  C_  ( p  vH  q )  ->  (
r  C_  A  ->  r  =  p ) ) ) )
711, 70sylanr1 652 . 2  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e. HAtoms  /\  q  C_  ( _|_ `  A
) ) )  -> 
( r  e. HAtoms  ->  ( r  C_  ( p  vH  q )  ->  (
r  C_  A  ->  r  =  p ) ) ) )
7271imp32 433 1  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. HAtoms  /\  q  C_  ( _|_ `  A ) ) )  /\  ( r  e. HAtoms  /\  r  C_  ( p  vH  q ) ) )  ->  ( r  C_  A  ->  r  =  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3322    C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   SHcsh 24281   CHcch 24282   _|_cort 24283    vH chj 24286   0Hc0h 24288    C_H ccm 24289  HAtomscat 24318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438  ax-hcompl 24555
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-cn 18806  df-cnp 18807  df-lm 18808  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cfil 20741  df-cau 20742  df-cmet 20743  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-ssp 24071  df-ph 24164  df-cbn 24215  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-hcau 24326  df-sh 24560  df-ch 24575  df-oc 24606  df-ch0 24607  df-shs 24662  df-span 24663  df-chj 24664  df-chsup 24665  df-pjh 24749  df-cm 24937  df-cv 25634  df-at 25693
This theorem is referenced by:  chirredlem4  25748
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