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Theorem mdslmd3i 23828
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mdslmd.4 . . . . . . . . . . 11  |-  D  e. 
CH
2 mdslmd.1 . . . . . . . . . . 11  |-  A  e. 
CH
3 chlej2 23006 . . . . . . . . . . . 12  |-  ( ( ( D  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  /\  D  C_  A )  ->  ( x  vH  D )  C_  (
x  vH  A )
)
43ex 424 . . . . . . . . . . 11  |-  ( ( D  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  ( D  C_  A  ->  (
x  vH  D )  C_  ( x  vH  A
) ) )
51, 2, 4mp3an12 1269 . . . . . . . . . 10  |-  ( x  e.  CH  ->  ( D  C_  A  ->  (
x  vH  D )  C_  ( x  vH  A
) ) )
65impcom 420 . . . . . . . . 9  |-  ( ( D  C_  A  /\  x  e.  CH )  ->  ( x  vH  D
)  C_  ( x  vH  A ) )
7 ssrin 3559 . . . . . . . . 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 16 . . . . . . . 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 695 . . . . . . 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 695 . . . . . 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 452 . . . . 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 3556 . . . . . . . 8  |-  ( ( x  C_  B  /\  x  C_  C )  <->  x  C_  ( B  i^i  C ) )
13 inass 3544 . . . . . . . . . . . . . 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 23791 . . . . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CH  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
1817ineq1d 3534 . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . 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 704 . . . . . . . . . . . 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 706 . . . . . . . . . . 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 22955 . . . . . . . . . . . . . . . 16  |-  ( A  i^i  B )  e. 
CH
24 mdi 23791 . . . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . 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 3544 . . . . . . . . . . . . . . 15  |-  ( ( A  i^i  B )  i^i  C )  =  ( A  i^i  ( B  i^i  C ) )
2827oveq2i 6085 . . . . . . . . . . . . . 14  |-  ( x  vH  ( ( A  i^i  B )  i^i 
C ) )  =  ( x  vH  ( A  i^i  ( B  i^i  C ) ) )
2926, 28syl6eq 2484 . . . . . . . . . . . . 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 703 . . . . . . . . . . . 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 705 . . . . . . . . . . 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 2468 . . . . . . . . . 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 440 . . . . . . . . 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 780 . . . . . . . 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 463 . . . . . . 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 700 . . . . . 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 3544 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  i^i  ( B  i^i  C ) )  =  ( A  i^i  ( C  i^i  ( B  i^i  C ) ) )
38 in12 3545 . . . . . . . . . . . . . 14  |-  ( C  i^i  ( B  i^i  C ) )  =  ( B  i^i  ( C  i^i  C ) )
39 inidm 3543 . . . . . . . . . . . . . . 15  |-  ( C  i^i  C )  =  C
4039ineq2i 3532 . . . . . . . . . . . . . 14  |-  ( B  i^i  ( C  i^i  C ) )  =  ( B  i^i  C )
4138, 40eqtri 2456 . . . . . . . . . . . . 13  |-  ( C  i^i  ( B  i^i  C ) )  =  ( B  i^i  C )
4241ineq2i 3532 . . . . . . . . . . . 12  |-  ( A  i^i  ( C  i^i  ( B  i^i  C ) ) )  =  ( A  i^i  ( B  i^i  C ) )
4337, 42eqtr2i 2457 . . . . . . . . . . 11  |-  ( A  i^i  ( B  i^i  C ) )  =  ( ( A  i^i  C
)  i^i  ( B  i^i  C ) )
44 ssrin 3559 . . . . . . . . . . 11  |-  ( ( 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 3385 . . . . . . . . . 10  |-  ( ( A  i^i  C ) 
C_  D  ->  ( A  i^i  ( B  i^i  C ) )  C_  ( D  i^i  ( B  i^i  C ) ) )
46 ssrin 3559 . . . . . . . . . 10  |-  ( D 
C_  A  ->  ( D  i^i  ( B  i^i  C ) )  C_  ( A  i^i  ( B  i^i  C ) ) )
4745, 46anim12i 550 . . . . . . . . 9  |-  ( ( ( 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 3356 . . . . . . . . 9  |-  ( ( 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 204 . . . . . . . 8  |-  ( ( ( A  i^i  C
)  C_  D  /\  D  C_  A )  -> 
( A  i^i  ( B  i^i  C ) )  =  ( D  i^i  ( B  i^i  C ) ) )
5049oveq2d 6090 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
5150ad3antlr 712 . . . . . 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 ) ) ) )
5236, 51eqtrd 2468 . . . . 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 ) ) ) )
5311, 52sseqtrd 3377 . . . 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
) ) ) )
5453ex 424 . . 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 ) ) ) ) )
5554ralrimiva 2782 . 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
) ) ) ) )
5614, 22chincli 22955 . . 3  |-  ( B  i^i  C )  e. 
CH
57 mdbr2 23792 . . 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
) ) ) ) ) )
581, 56, 57mp2an 654 . 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 ) ) ) ) )
5955, 58sylibr 204 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 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2698    i^i cin 3312    C_ wss 3313   class class class wbr 4205  (class class class)co 6074   CHcch 22425    vH chj 22429    MH cmd 22462
This theorem is referenced by:  mdslmd4i  23829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cc 8308  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062  ax-mulf 9063  ax-hilex 22495  ax-hfvadd 22496  ax-hvcom 22497  ax-hvass 22498  ax-hv0cl 22499  ax-hvaddid 22500  ax-hfvmul 22501  ax-hvmulid 22502  ax-hvmulass 22503  ax-hvdistr1 22504  ax-hvdistr2 22505  ax-hvmul0 22506  ax-hfi 22574  ax-his1 22577  ax-his2 22578  ax-his3 22579  ax-his4 22580  ax-hcompl 22697
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-of 6298  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-2o 6718  df-oadd 6721  df-omul 6722  df-er 6898  df-map 7013  df-pm 7014  df-ixp 7057  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-fi 7409  df-sup 7439  df-oi 7472  df-card 7819  df-acn 7822  df-cda 8041  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-7 10056  df-8 10057  df-9 10058  df-10 10059  df-n0 10215  df-z 10276  df-dec 10376  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ioo 10913  df-ico 10915  df-icc 10916  df-fz 11037  df-fzo 11129  df-fl 11195  df-seq 11317  df-exp 11376  df-hash 11612  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-rlim 12276  df-sum 12473  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-starv 13537  df-sca 13538  df-vsca 13539  df-tset 13541  df-ple 13542  df-ds 13544  df-unif 13545  df-hom 13546  df-cco 13547  df-rest 13643  df-topn 13644  df-topgen 13660  df-pt 13661  df-prds 13664  df-xrs 13719  df-0g 13720  df-gsum 13721  df-qtop 13726  df-imas 13727  df-xps 13729  df-mre 13804  df-mrc 13805  df-acs 13807  df-mnd 14683  df-submnd 14732  df-mulg 14808  df-cntz 15109  df-cmn 15407  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-fbas 16692  df-fg 16693  df-cnfld 16697  df-top 16956  df-bases 16958  df-topon 16959  df-topsp 16960  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-cn 17284  df-cnp 17285  df-lm 17286  df-haus 17372  df-tx 17587  df-hmeo 17780  df-fil 17871  df-fm 17963  df-flim 17964  df-flf 17965  df-xms 18343  df-ms 18344  df-tms 18345  df-cfil 19201  df-cau 19202  df-cmet 19203  df-grpo 21772  df-gid 21773  df-ginv 21774  df-gdiv 21775  df-ablo 21863  df-subgo 21883  df-vc 22018  df-nv 22064  df-va 22067  df-ba 22068  df-sm 22069  df-0v 22070  df-vs 22071  df-nmcv 22072  df-ims 22073  df-dip 22190  df-ssp 22214  df-ph 22307  df-cbn 22358  df-hnorm 22464  df-hba 22465  df-hvsub 22467  df-hlim 22468  df-hcau 22469  df-sh 22702  df-ch 22717  df-oc 22747  df-ch0 22748  df-shs 22803  df-chj 22805  df-md 23776
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