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Theorem cm2j 27259
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 27253 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  C_H  A ) )
2 cmbr 27223 . . . . . . . . . . . 12  |-  ( ( B  e.  CH  /\  A  e.  CH )  ->  ( B  C_H  A  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
32ancoms 454 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( B  C_H  A  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
41, 3bitrd 256 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  =  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) ) ) )
54biimpa 486 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_H  B )  ->  B  =  ( ( B  i^i  A
)  vH  ( B  i^i  ( _|_ `  A
) ) ) )
6 incom 3655 . . . . . . . . . 10  |-  ( B  i^i  A )  =  ( A  i^i  B
)
7 incom 3655 . . . . . . . . . 10  |-  ( B  i^i  ( _|_ `  A
) )  =  ( ( _|_ `  A
)  i^i  B )
86, 7oveq12i 6314 . . . . . . . . 9  |-  ( ( B  i^i  A )  vH  ( B  i^i  ( _|_ `  A ) ) )  =  ( ( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) )
95, 8syl6eq 2479 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_H  B )  ->  B  =  ( ( A  i^i  B
)  vH  ( ( _|_ `  A )  i^i 
B ) ) )
1093adantl3 1163 . . . . . . 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 721 . . . . . 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 27253 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  C  C_H  A ) )
13 cmbr 27223 . . . . . . . . . . . 12  |-  ( ( C  e.  CH  /\  A  e.  CH )  ->  ( C  C_H  A  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1413ancoms 454 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( C  C_H  A  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1512, 14bitrd 256 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  C  =  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) ) ) )
1615biimpa 486 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  C  =  ( ( C  i^i  A
)  vH  ( C  i^i  ( _|_ `  A
) ) ) )
17 incom 3655 . . . . . . . . . 10  |-  ( C  i^i  A )  =  ( A  i^i  C
)
18 incom 3655 . . . . . . . . . 10  |-  ( C  i^i  ( _|_ `  A
) )  =  ( ( _|_ `  A
)  i^i  C )
1917, 18oveq12i 6314 . . . . . . . . 9  |-  ( ( C  i^i  A )  vH  ( C  i^i  ( _|_ `  A ) ) )  =  ( ( A  i^i  C
)  vH  ( ( _|_ `  A )  i^i 
C ) )
2016, 19syl6eq 2479 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  C  =  ( ( A  i^i  C
)  vH  ( ( _|_ `  A )  i^i 
C ) ) )
21203adantl2 1162 . . . . . . 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 720 . . . . . 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 6320 . . . . 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 27138 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
25 choccl 26945 . . . . . . . . . 10  |-  ( A  e.  CH  ->  ( _|_ `  A )  e. 
CH )
26 chincl 27138 . . . . . . . . . 10  |-  ( ( ( _|_ `  A
)  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A
)  i^i  B )  e.  CH )
2725, 26sylan 473 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( ( _|_ `  A
)  i^i  B )  e.  CH )
2824, 27jca 534 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( ( A  i^i  B )  e.  CH  /\  ( ( _|_ `  A
)  i^i  B )  e.  CH ) )
29 chincl 27138 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
30 chincl 27138 . . . . . . . . . 10  |-  ( ( ( _|_ `  A
)  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  A
)  i^i  C )  e.  CH )
3125, 30sylan 473 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  A
)  i^i  C )  e.  CH )
3229, 31jca 534 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  C )  e.  CH  /\  ( ( _|_ `  A
)  i^i  C )  e.  CH ) )
33 chj4 27174 . . . . . . . 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 479 . . . . . . 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 1319 . . . . . 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 466 . . . . 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 3655 . . . . . . 7  |-  ( A  i^i  ( B  vH  C ) )  =  ( ( B  vH  C )  i^i  A
)
38 fh1 27257 . . . . . . 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 2478 . . . . . 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 3655 . . . . . . 7  |-  ( ( _|_ `  A )  i^i  ( B  vH  C ) )  =  ( ( B  vH  C )  i^i  ( _|_ `  A ) )
41253anim1i 1191 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  A
)  e.  CH  /\  B  e.  CH  /\  C  e.  CH ) )
4241adantr 466 . . . . . . . 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 27254 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B ) )
44433adant3 1025 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  B  <->  ( _|_ `  A )  C_H  B
) )
45 cmcm3 27254 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  ( _|_ `  A )  C_H  C ) )
46453adant2 1024 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  ( _|_ `  A )  C_H  C
) )
4744, 46anbi12d 715 . . . . . . . . 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 486 . . . . . . . 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 27257 . . . . . . . 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 665 . . . . . . 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 2478 . . . . . 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 6320 . . . . 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 2467 . . . 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 435 . . 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 26996 . . . . 5  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
56 cmcm 27253 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  C_H  ( B  vH  C )  <->  ( B  vH  C )  C_H  A
) )
57 cmbr 27223 . . . . . . 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 454 . . . . . 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 256 . . . . 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 476 . . . 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 1201 . . 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 237 . 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 430 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    i^i cin 3435   class class class wbr 4420   ` cfv 5598  (class class class)co 6302   CHcch 26568   _|_cort 26569    vH chj 26572    C_H ccm 26575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-inf2 8149  ax-cc 8866  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617  ax-pre-sup 9618  ax-addf 9619  ax-mulf 9620  ax-hilex 26638  ax-hfvadd 26639  ax-hvcom 26640  ax-hvass 26641  ax-hv0cl 26642  ax-hvaddid 26643  ax-hfvmul 26644  ax-hvmulid 26645  ax-hvmulass 26646  ax-hvdistr1 26647  ax-hvdistr2 26648  ax-hvmul0 26649  ax-hfi 26718  ax-his1 26721  ax-his2 26722  ax-his3 26723  ax-his4 26724  ax-hcompl 26841
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-se 4810  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-isom 5607  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-supp 6923  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7887  df-fi 7928  df-sup 7959  df-inf 7960  df-oi 8028  df-card 8375  df-acn 8378  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-div 10271  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-uz 11161  df-q 11266  df-rp 11304  df-xneg 11410  df-xadd 11411  df-xmul 11412  df-ioo 11640  df-ico 11642  df-icc 11643  df-fz 11786  df-fzo 11917  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13151  df-re 13152  df-im 13153  df-sqrt 13287  df-abs 13288  df-clim 13540  df-rlim 13541  df-sum 13741  df-struct 15111  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-plusg 15191  df-mulr 15192  df-starv 15193  df-sca 15194  df-vsca 15195  df-ip 15196  df-tset 15197  df-ple 15198  df-ds 15200  df-unif 15201  df-hom 15202  df-cco 15203  df-rest 15309  df-topn 15310  df-0g 15328  df-gsum 15329  df-topgen 15330  df-pt 15331  df-prds 15334  df-xrs 15388  df-qtop 15394  df-imas 15395  df-xps 15398  df-mre 15480  df-mrc 15481  df-acs 15483  df-mgm 16476  df-sgrp 16515  df-mnd 16525  df-submnd 16571  df-mulg 16664  df-cntz 16959  df-cmn 17420  df-psmet 18950  df-xmet 18951  df-met 18952  df-bl 18953  df-mopn 18954  df-fbas 18955  df-fg 18956  df-cnfld 18959  df-top 19908  df-bases 19909  df-topon 19910  df-topsp 19911  df-cld 20021  df-ntr 20022  df-cls 20023  df-nei 20101  df-cn 20230  df-cnp 20231  df-lm 20232  df-haus 20318  df-tx 20564  df-hmeo 20757  df-fil 20848  df-fm 20940  df-flim 20941  df-flf 20942  df-xms 21322  df-ms 21323  df-tms 21324  df-cfil 22212  df-cau 22213  df-cmet 22214  df-grpo 25905  df-gid 25906  df-ginv 25907  df-gdiv 25908  df-ablo 25996  df-subgo 26016  df-vc 26151  df-nv 26197  df-va 26200  df-ba 26201  df-sm 26202  df-0v 26203  df-vs 26204  df-nmcv 26205  df-ims 26206  df-dip 26323  df-ssp 26347  df-ph 26440  df-cbn 26491  df-hnorm 26607  df-hba 26608  df-hvsub 26610  df-hlim 26611  df-hcau 26612  df-sh 26846  df-ch 26860  df-oc 26891  df-ch0 26892  df-shs 26947  df-chj 26949  df-cm 27222
This theorem is referenced by:  cm2ji  27264
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