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Theorem cm2j 25174
Description: A lattice element that commutes with two others also commutes with their join. Theorem 4.2 of [Beran] p. 49. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
cm2j  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  A  C_H  ( B  vH  C ) )

Proof of Theorem cm2j
StepHypRef Expression
1 cmcm 25168 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  C_H  A ) )
2 cmbr 25138 . . . . . . . . . . . 12  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_H  A  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
32ancoms 453 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( B  C_H  A  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
41, 3bitrd 253 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
54biimpa 484 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_H  B )  ->  B  =  ( ( B  i^i  A
)  vH  ( B  i^i  ( _|_ `  A
) ) ) )
6 incom 3650 . . . . . . . . . 10  |-  ( B  i^i  A )  =  ( A  i^i  B
)
7 incom 3650 . . . . . . . . . 10  |-  ( B  i^i  ( _|_ `  A
) )  =  ( ( _|_ `  A
)  i^i  B )
86, 7oveq12i 6211 . . . . . . . . 9  |-  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) )  =  ( ( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) )
95, 8syl6eq 2511 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_H  B )  ->  B  =  ( ( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) ) )
1093adantl3 1146 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_H  B )  ->  B  =  ( ( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) ) )
1110adantrr 716 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  B  =  ( ( A  i^i  B )  vH  ( ( _|_ `  A )  i^i  B ) ) )
12 cmcm 25168 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  C  C_H  A ) )
13 cmbr 25138 . . . . . . . . . . . 12  |-  ( ( C  e.  CH  /\  A  e.  CH )  ->  ( C  C_H  A  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1413ancoms 453 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( C  C_H  A  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1512, 14bitrd 253 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1615biimpa 484 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  C  =  ( ( C  i^i  A
)  vH  ( C  i^i  ( _|_ `  A
) ) ) )
17 incom 3650 . . . . . . . . . 10  |-  ( C  i^i  A )  =  ( A  i^i  C
)
18 incom 3650 . . . . . . . . . 10  |-  ( C  i^i  ( _|_ `  A
) )  =  ( ( _|_ `  A
)  i^i  C )
1917, 18oveq12i 6211 . . . . . . . . 9  |-  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) )  =  ( ( A  i^i  C
)  vH  ( ( _|_ `  A )  i^i 
C ) )
2016, 19syl6eq 2511 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  C  =  ( ( A  i^i  C
)  vH  ( ( _|_ `  A )  i^i 
C ) ) )
21203adantl2 1145 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  C  =  ( ( A  i^i  C
)  vH  ( ( _|_ `  A )  i^i 
C ) ) )
2221adantrl 715 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  C  =  ( ( A  i^i  C )  vH  ( ( _|_ `  A )  i^i  C ) ) )
2311, 22oveq12d 6217 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( B  vH  C )  =  ( ( ( A  i^i  B )  vH  ( ( _|_ `  A )  i^i  B ) )  vH  ( ( A  i^i  C )  vH  ( ( _|_ `  A
)  i^i  C )
) ) )
24 chincl 25053 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
25 choccl 24860 . . . . . . . . . 10  |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
CH )
26 chincl 25053 . . . . . . . . . 10  |-  ( ( ( _|_ `  A
)  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A
)  i^i  B )  e.  CH )
2725, 26sylan 471 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A
)  i^i  B )  e.  CH )
2824, 27jca 532 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( ( A  i^i  B )  e.  CH  /\  ( ( _|_ `  A
)  i^i  B )  e.  CH ) )
29 chincl 25053 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
30 chincl 25053 . . . . . . . . . 10  |-  ( ( ( _|_ `  A
)  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  A
)  i^i  C )  e.  CH )
3125, 30sylan 471 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  A
)  i^i  C )  e.  CH )
3229, 31jca 532 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  C )  e.  CH  /\  ( ( _|_ `  A
)  i^i  C )  e.  CH ) )
33 chj4 25089 . . . . . . . 8  |-  ( ( ( ( A  i^i  B )  e.  CH  /\  ( ( _|_ `  A
)  i^i  B )  e.  CH )  /\  (
( A  i^i  C
)  e.  CH  /\  ( ( _|_ `  A
)  i^i  C )  e.  CH ) )  -> 
( ( ( A  i^i  B )  vH  ( ( _|_ `  A
)  i^i  B )
)  vH  ( ( A  i^i  C )  vH  ( ( _|_ `  A
)  i^i  C )
) )  =  ( ( ( A  i^i  B )  vH  ( A  i^i  C ) )  vH  ( ( ( _|_ `  A )  i^i  B )  vH  ( ( _|_ `  A
)  i^i  C )
) ) )
3428, 32, 33syl2an 477 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( (
( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) )  vH  ( ( A  i^i  C )  vH  ( ( _|_ `  A )  i^i  C ) ) )  =  ( ( ( A  i^i  B
)  vH  ( A  i^i  C ) )  vH  ( ( ( _|_ `  A )  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
C ) ) ) )
35343impdi 1274 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( ( A  i^i  B )  vH  ( ( _|_ `  A )  i^i  B ) )  vH  ( ( A  i^i  C )  vH  ( ( _|_ `  A
)  i^i  C )
) )  =  ( ( ( A  i^i  B )  vH  ( A  i^i  C ) )  vH  ( ( ( _|_ `  A )  i^i  B )  vH  ( ( _|_ `  A
)  i^i  C )
) ) )
3635adantr 465 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) )  vH  ( ( A  i^i  C )  vH  ( ( _|_ `  A )  i^i  C ) ) )  =  ( ( ( A  i^i  B
)  vH  ( A  i^i  C ) )  vH  ( ( ( _|_ `  A )  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
C ) ) ) )
37 incom 3650 . . . . . . 7  |-  ( A  i^i  ( B  vH  C ) )  =  ( ( B  vH  C )  i^i  A
)
38 fh1 25172 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( A  i^i  B
)  vH  ( A  i^i  C ) ) )
3937, 38syl5reqr 2510 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  =  ( ( B  vH  C )  i^i  A ) )
40 incom 3650 . . . . . . 7  |-  ( ( _|_ `  A )  i^i  ( B  vH  C ) )  =  ( ( B  vH  C )  i^i  ( _|_ `  A ) )
41253anim1i 1174 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  A
)  e.  CH  /\  B  e.  CH  /\  C  e.  CH ) )
4241adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( _|_ `  A )  e. 
CH  /\  B  e.  CH 
/\  C  e.  CH ) )
43 cmcm3 25169 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B ) )
44433adant3 1008 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B
) )
45 cmcm3 25169 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  ( _|_ `  A )  C_H  C ) )
46453adant2 1007 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  ( _|_ `  A )  C_H  C
) )
4744, 46anbi12d 710 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C_H  B  /\  A  C_H  C )  <-> 
( ( _|_ `  A
)  C_H  B  /\  ( _|_ `  A )  C_H  C ) ) )
4847biimpa 484 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( _|_ `  A )  C_H 
B  /\  ( _|_ `  A )  C_H  C
) )
49 fh1 25172 . . . . . . . 8  |-  ( ( ( ( _|_ `  A
)  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  (
( _|_ `  A
)  C_H  B  /\  ( _|_ `  A )  C_H  C ) )  ->  ( ( _|_ `  A )  i^i  ( B  vH  C ) )  =  ( ( ( _|_ `  A )  i^i  B )  vH  ( ( _|_ `  A
)  i^i  C )
) )
5042, 48, 49syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( _|_ `  A )  i^i  ( B  vH  C
) )  =  ( ( ( _|_ `  A
)  i^i  B )  vH  ( ( _|_ `  A
)  i^i  C )
) )
5140, 50syl5reqr 2510 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( _|_ `  A
)  i^i  B )  vH  ( ( _|_ `  A
)  i^i  C )
)  =  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) )
5239, 51oveq12d 6217 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( A  i^i  B
)  vH  ( A  i^i  C ) )  vH  ( ( ( _|_ `  A )  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
C ) ) )  =  ( ( ( B  vH  C )  i^i  A )  vH  ( ( B  vH  C )  i^i  ( _|_ `  A ) ) ) )
5323, 36, 523eqtrd 2499 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A
)  vH  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) ) )
5453ex 434 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C_H  B  /\  A  C_H  C )  ->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A
)  vH  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) ) ) )
55 chjcl 24911 . . . . 5  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
56 cmcm 25168 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  C_H  ( B  vH  C )  <->  ( B  vH  C )  C_H  A
) )
57 cmbr 25138 . . . . . . 7  |-  ( ( ( B  vH  C
)  e.  CH  /\  A  e.  CH )  ->  ( ( B  vH  C )  C_H  A  <->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A )  vH  ( ( B  vH  C )  i^i  ( _|_ `  A ) ) ) ) )
5857ancoms 453 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( ( B  vH  C )  C_H  A  <->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A )  vH  ( ( B  vH  C )  i^i  ( _|_ `  A ) ) ) ) )
5956, 58bitrd 253 . . . . 5  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  C_H  ( B  vH  C )  <->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A
)  vH  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) ) ) )
6055, 59sylan2 474 . . . 4  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  C_H  ( B  vH  C
)  <->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A
)  vH  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) ) ) )
61603impb 1184 . . 3  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  ( B  vH  C )  <->  ( B  vH  C )  =  ( ( ( B  vH  C )  i^i  A
)  vH  ( ( B  vH  C )  i^i  ( _|_ `  A
) ) ) ) )
6254, 61sylibrd 234 . 2  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C_H  B  /\  A  C_H  C )  ->  A  C_H  ( B  vH  C ) ) )
6362imp 429 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  A  C_H  ( B  vH  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3434   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   CHcch 24482   _|_cort 24483    vH chj 24486    C_H ccm 24489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cc 8714  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472  ax-hilex 24552  ax-hfvadd 24553  ax-hvcom 24554  ax-hvass 24555  ax-hv0cl 24556  ax-hvaddid 24557  ax-hfvmul 24558  ax-hvmulid 24559  ax-hvmulass 24560  ax-hvdistr1 24561  ax-hvdistr2 24562  ax-hvmul0 24563  ax-hfi 24632  ax-his1 24635  ax-his2 24636  ax-his3 24637  ax-his4 24638  ax-hcompl 24755
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-rlim 13084  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-cnp 18963  df-lm 18964  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cfil 20897  df-cau 20898  df-cmet 20899  df-grpo 23829  df-gid 23830  df-ginv 23831  df-gdiv 23832  df-ablo 23920  df-subgo 23940  df-vc 24075  df-nv 24121  df-va 24124  df-ba 24125  df-sm 24126  df-0v 24127  df-vs 24128  df-nmcv 24129  df-ims 24130  df-dip 24247  df-ssp 24271  df-ph 24364  df-cbn 24415  df-hnorm 24521  df-hba 24522  df-hvsub 24524  df-hlim 24525  df-hcau 24526  df-sh 24760  df-ch 24775  df-oc 24806  df-ch0 24807  df-shs 24862  df-chj 24864  df-cm 25137
This theorem is referenced by:  cm2ji  25179
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