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Theorem mdslmd3i 22904
Description: Modular pair conditions that imply the modular pair property in a sublattice. Lemma 1.5.1 of [MaedaMaeda] p. 2. (Contributed by NM, 23-Dec-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslmd.1  |-  A  e. 
CH
mdslmd.2  |-  B  e. 
CH
mdslmd.3  |-  C  e. 
CH
mdslmd.4  |-  D  e. 
CH
Assertion
Ref Expression
mdslmd3i  |-  ( ( ( A  MH  B  /\  ( A  i^i  B
)  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  ->  D  MH  ( B  i^i  C
) )

Proof of Theorem mdslmd3i
StepHypRef Expression
1 mdslmd.4 . . . . . . . . . . 11  |-  D  e. 
CH
2 mdslmd.1 . . . . . . . . . . 11  |-  A  e. 
CH
3 chlej2 22082 . . . . . . . . . . . 12  |-  ( ( ( D  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  D  C_  A )  ->  ( x  vH  D )  C_  (
x  vH  A )
)
43ex 425 . . . . . . . . . . 11  |-  ( ( D  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  ( D  C_  A  ->  (
x  vH  D )  C_  ( x  vH  A
) ) )
51, 2, 4mp3an12 1272 . . . . . . . . . 10  |-  ( x  e.  CH  ->  ( D  C_  A  ->  (
x  vH  D )  C_  ( x  vH  A
) ) )
65impcom 421 . . . . . . . . 9  |-  ( ( D  C_  A  /\  x  e.  CH )  ->  ( x  vH  D
)  C_  ( x  vH  A ) )
7 ssrin 3395 . . . . . . . . 9  |-  ( ( x  vH  D ) 
C_  ( x  vH  A )  ->  (
( x  vH  D
)  i^i  ( B  i^i  C ) )  C_  ( ( x  vH  A )  i^i  ( B  i^i  C ) ) )
86, 7syl 17 . . . . . . . 8  |-  ( ( D  C_  A  /\  x  e.  CH )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C ) ) 
C_  ( ( x  vH  A )  i^i  ( B  i^i  C
) ) )
98adantll 697 . . . . . . 7  |-  ( ( ( ( A  i^i  C )  C_  D  /\  D  C_  A )  /\  x  e.  CH )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C ) ) 
C_  ( ( x  vH  A )  i^i  ( B  i^i  C
) ) )
109adantll 697 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  /\  x  e.  CH )  ->  (
( x  vH  D
)  i^i  ( B  i^i  C ) )  C_  ( ( x  vH  A )  i^i  ( B  i^i  C ) ) )
1110adantr 453 . . . . 5  |-  ( ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  (
( A  i^i  C
)  C_  D  /\  D  C_  A ) )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C
) )  C_  (
( x  vH  A
)  i^i  ( B  i^i  C ) ) )
12 ssin 3392 . . . . . . . 8  |-  ( ( x  C_  B  /\  x  C_  C )  <->  x  C_  ( B  i^i  C ) )
13 inass 3380 . . . . . . . . . . . . . 14  |-  ( ( ( x  vH  A
)  i^i  B )  i^i  C )  =  ( ( x  vH  A
)  i^i  ( B  i^i  C ) )
14 mdslmd.2 . . . . . . . . . . . . . . . 16  |-  B  e. 
CH
15 mdi 22867 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  x  e.  CH )  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
162, 15mp3anl1 1276 . . . . . . . . . . . . . . . 16  |-  ( ( ( B  e.  CH  /\  x  e.  CH )  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
1714, 16mpanl1 664 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CH  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
1817ineq1d 3370 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
( x  vH  A
)  i^i  B )  i^i  C )  =  ( ( x  vH  ( A  i^i  B ) )  i^i  C ) )
1913, 18syl5eqr 2330 . . . . . . . . . . . . 13  |-  ( ( x  e.  CH  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  ( B  i^i  C
) )  =  ( ( x  vH  ( A  i^i  B ) )  i^i  C ) )
2019adantrlr 706 . . . . . . . . . . . 12  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  x  C_  B
) )  ->  (
( x  vH  A
)  i^i  ( B  i^i  C ) )  =  ( ( x  vH  ( A  i^i  B ) )  i^i  C ) )
2120adantrrr 708 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( x  C_  B  /\  x  C_  C
) ) )  -> 
( ( x  vH  A )  i^i  ( B  i^i  C ) )  =  ( ( x  vH  ( A  i^i  B ) )  i^i  C
) )
22 mdslmd.3 . . . . . . . . . . . . . . 15  |-  C  e. 
CH
232, 14chincli 22031 . . . . . . . . . . . . . . . 16  |-  ( A  i^i  B )  e. 
CH
24 mdi 22867 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  i^i  B )  e.  CH  /\  C  e.  CH  /\  x  e.  CH )  /\  (
( A  i^i  B
)  MH  C  /\  x  C_  C ) )  ->  ( ( x  vH  ( A  i^i  B ) )  i^i  C
)  =  ( x  vH  ( ( A  i^i  B )  i^i 
C ) ) )
2523, 24mp3anl1 1276 . . . . . . . . . . . . . . 15  |-  ( ( ( C  e.  CH  /\  x  e.  CH )  /\  ( ( A  i^i  B )  MH  C  /\  x  C_  C ) )  ->  ( ( x  vH  ( A  i^i  B ) )  i^i  C
)  =  ( x  vH  ( ( A  i^i  B )  i^i 
C ) ) )
2622, 25mpanl1 664 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( ( A  i^i  B )  MH  C  /\  x  C_  C ) )  ->  ( ( x  vH  ( A  i^i  B ) )  i^i  C
)  =  ( x  vH  ( ( A  i^i  B )  i^i 
C ) ) )
27 inass 3380 . . . . . . . . . . . . . . 15  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2827oveq2i 5830 . . . . . . . . . . . . . 14  |-  ( x  vH  ( ( A  i^i  B )  i^i 
C ) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) )
2926, 28syl6eq 2332 . . . . . . . . . . . . 13  |-  ( ( x  e.  CH  /\  ( ( A  i^i  B )  MH  C  /\  x  C_  C ) )  ->  ( ( x  vH  ( A  i^i  B ) )  i^i  C
)  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3029adantrll 705 . . . . . . . . . . . 12  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  x  C_  C
) )  ->  (
( x  vH  ( A  i^i  B ) )  i^i  C )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3130adantrrl 707 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( x  C_  B  /\  x  C_  C
) ) )  -> 
( ( x  vH  ( A  i^i  B ) )  i^i  C )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3221, 31eqtrd 2316 . . . . . . . . . 10  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( x  C_  B  /\  x  C_  C
) ) )  -> 
( ( x  vH  A )  i^i  ( B  i^i  C ) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3332ancoms 441 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( x  C_  B  /\  x  C_  C
) )  /\  x  e.  CH )  ->  (
( x  vH  A
)  i^i  ( B  i^i  C ) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3433an32s 782 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  x  e.  CH )  /\  ( x  C_  B  /\  x  C_  C
) )  ->  (
( x  vH  A
)  i^i  ( B  i^i  C ) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3512, 34sylan2br 464 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( ( x  vH  A )  i^i  ( B  i^i  C
) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
3635adantllr 702 . . . . . 6  |-  ( ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  (
( A  i^i  C
)  C_  D  /\  D  C_  A ) )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( ( x  vH  A )  i^i  ( B  i^i  C
) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) ) )
37 inass 3380 . . . . . . . . . . . . 13  |-  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  C ) ) )
38 in12 3381 . . . . . . . . . . . . . . 15  |-  ( C  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( C  i^i  C ) )
39 inidm 3379 . . . . . . . . . . . . . . . 16  |-  ( C  i^i  C )  =  C
4039ineq2i 3368 . . . . . . . . . . . . . . 15  |-  ( B  i^i  ( C  i^i  C ) )  =  ( B  i^i  C )
4138, 40eqtri 2304 . . . . . . . . . . . . . 14  |-  ( C  i^i  ( B  i^i  C ) )  =  ( B  i^i  C )
4241ineq2i 3368 . . . . . . . . . . . . 13  |-  ( A  i^i  ( C  i^i  ( B  i^i  C ) ) )  =  ( A  i^i  ( B  i^i  C ) )
4337, 42eqtr2i 2305 . . . . . . . . . . . 12  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  C ) )
44 ssrin 3395 . . . . . . . . . . . 12  |-  ( ( A  i^i  C ) 
C_  D  ->  (
( A  i^i  C
)  i^i  ( B  i^i  C ) )  C_  ( D  i^i  ( B  i^i  C ) ) )
4543, 44syl5eqss 3223 . . . . . . . . . . 11  |-  ( ( A  i^i  C ) 
C_  D  ->  ( A  i^i  ( B  i^i  C ) )  C_  ( D  i^i  ( B  i^i  C ) ) )
46 ssrin 3395 . . . . . . . . . . 11  |-  ( D 
C_  A  ->  ( D  i^i  ( B  i^i  C ) )  C_  ( A  i^i  ( B  i^i  C ) ) )
4745, 46anim12i 551 . . . . . . . . . 10  |-  ( ( ( A  i^i  C
)  C_  D  /\  D  C_  A )  -> 
( ( A  i^i  ( B  i^i  C ) )  C_  ( D  i^i  ( B  i^i  C
) )  /\  ( D  i^i  ( B  i^i  C ) )  C_  ( A  i^i  ( B  i^i  C ) ) ) )
48 eqss 3195 . . . . . . . . . 10  |-  ( ( A  i^i  ( B  i^i  C ) )  =  ( D  i^i  ( B  i^i  C ) )  <->  ( ( A  i^i  ( B  i^i  C ) )  C_  ( D  i^i  ( B  i^i  C ) )  /\  ( D  i^i  ( B  i^i  C ) )  C_  ( A  i^i  ( B  i^i  C ) ) ) )
4947, 48sylibr 205 . . . . . . . . 9  |-  ( ( ( A  i^i  C
)  C_  D  /\  D  C_  A )  -> 
( A  i^i  ( B  i^i  C ) )  =  ( D  i^i  ( B  i^i  C ) ) )
5049oveq2d 5835 . . . . . . . 8  |-  ( ( ( A  i^i  C
)  C_  D  /\  D  C_  A )  -> 
( x  vH  ( A  i^i  ( B  i^i  C ) ) )  =  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) )
5150adantl 454 . . . . . . 7  |-  ( ( ( A  MH  B  /\  ( A  i^i  B
)  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  ->  (
x  vH  ( A  i^i  ( B  i^i  C
) ) )  =  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) )
5251ad2antrr 709 . . . . . 6  |-  ( ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  (
( A  i^i  C
)  C_  D  /\  D  C_  A ) )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( x  vH  ( A  i^i  ( B  i^i  C ) ) )  =  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) )
5336, 52eqtrd 2316 . . . . 5  |-  ( ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  (
( A  i^i  C
)  C_  D  /\  D  C_  A ) )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( ( x  vH  A )  i^i  ( B  i^i  C
) )  =  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) )
5411, 53sseqtrd 3215 . . . 4  |-  ( ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  (
( A  i^i  C
)  C_  D  /\  D  C_  A ) )  /\  x  e.  CH )  /\  x  C_  ( B  i^i  C ) )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C
) )  C_  (
x  vH  ( D  i^i  ( B  i^i  C
) ) ) )
5554ex 425 . . 3  |-  ( ( ( ( A  MH  B  /\  ( A  i^i  B )  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  /\  x  e.  CH )  ->  (
x  C_  ( B  i^i  C )  ->  (
( x  vH  D
)  i^i  ( B  i^i  C ) )  C_  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) ) )
5655ralrimiva 2627 . 2  |-  ( ( ( A  MH  B  /\  ( A  i^i  B
)  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  ->  A. x  e.  CH  ( x  C_  ( B  i^i  C )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C
) )  C_  (
x  vH  ( D  i^i  ( B  i^i  C
) ) ) ) )
5714, 22chincli 22031 . . 3  |-  ( B  i^i  C )  e. 
CH
58 mdbr2 22868 . . 3  |-  ( ( D  e.  CH  /\  ( B  i^i  C )  e.  CH )  -> 
( D  MH  ( B  i^i  C )  <->  A. x  e.  CH  ( x  C_  ( B  i^i  C )  ->  ( ( x  vH  D )  i^i  ( B  i^i  C
) )  C_  (
x  vH  ( D  i^i  ( B  i^i  C
) ) ) ) ) )
591, 57, 58mp2an 656 . 2  |-  ( D  MH  ( B  i^i  C )  <->  A. x  e.  CH  ( x  C_  ( B  i^i  C )  -> 
( ( x  vH  D )  i^i  ( B  i^i  C ) ) 
C_  ( x  vH  ( D  i^i  ( B  i^i  C ) ) ) ) )
6056, 59sylibr 205 1  |-  ( ( ( A  MH  B  /\  ( A  i^i  B
)  MH  C )  /\  ( ( A  i^i  C )  C_  D  /\  D  C_  A
) )  ->  D  MH  ( B  i^i  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688   A.wral 2544    i^i cin 3152    C_ wss 3153   class class class wbr 4024  (class class class)co 5819   CHcch 21501    vH chj 21505    MH cmd 21538
This theorem is referenced by:  mdslmd4i  22905
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cc 8056  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812  ax-hilex 21571  ax-hfvadd 21572  ax-hvcom 21573  ax-hvass 21574  ax-hv0cl 21575  ax-hvaddid 21576  ax-hfvmul 21577  ax-hvmulid 21578  ax-hvmulass 21579  ax-hvdistr1 21580  ax-hvdistr2 21581  ax-hvmul0 21582  ax-hfi 21650  ax-his1 21653  ax-his2 21654  ax-his3 21655  ax-his4 21656  ax-hcompl 21773
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-acn 7570  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-rlim 11957  df-sum 12153  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-cn 16951  df-cnp 16952  df-lm 16953  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cfil 18675  df-cau 18676  df-cmet 18677  df-grpo 20850  df-gid 20851  df-ginv 20852  df-gdiv 20853  df-ablo 20941  df-subgo 20961  df-vc 21094  df-nv 21140  df-va 21143  df-ba 21144  df-sm 21145  df-0v 21146  df-vs 21147  df-nmcv 21148  df-ims 21149  df-dip 21266  df-ssp 21290  df-ph 21383  df-cbn 21434  df-hnorm 21540  df-hba 21541  df-hvsub 21543  df-hlim 21544  df-hcau 21545  df-sh 21778  df-ch 21793  df-oc 21823  df-ch0 21824  df-shs 21879  df-chj 21881  df-md 22852
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