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Theorem chirredlem2 23895
Description: Lemma for chirredi 23898. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
chirred.1  |-  A  e. 
CH
Assertion
Ref Expression
chirredlem2  |-  ( ( ( ( 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 )
Distinct variable group:    q, p, r, A

Proof of Theorem chirredlem2
StepHypRef Expression
1 atelch 23848 . . . . . 6  |-  ( p  e. HAtoms  ->  p  e.  CH )
2 chjcom 23009 . . . . . 6  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( p  vH  q
)  =  ( q  vH  p ) )
31, 2sylan 459 . . . . 5  |-  ( ( p  e. HAtoms  /\  q  e.  CH )  ->  (
p  vH  q )  =  ( q  vH  p ) )
43ad2ant2r 729 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( p  vH  q
)  =  ( q  vH  p ) )
54adantr 453 . . 3  |-  ( ( ( ( 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  q
)  =  ( q  vH  p ) )
65ineq2d 3543 . 2  |-  ( ( ( ( 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 ) )  =  ( ( _|_ `  r
)  i^i  ( q  vH  p ) ) )
7 atelch 23848 . . . . . . . . . . 11  |-  ( r  e. HAtoms  ->  r  e.  CH )
8 choccl 22809 . . . . . . . . . . 11  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
97, 8syl 16 . . . . . . . . . 10  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
10 id 21 . . . . . . . . . 10  |-  ( q  e.  CH  ->  q  e.  CH )
119, 10, 13anim123i 1140 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  q  e.  CH  /\  p  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
12113com13 1159 . . . . . . . 8  |-  ( ( p  e. HAtoms  /\  q  e.  CH  /\  r  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
13123expa 1154 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
1413adantllr 701 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH ) )
1514adantlrr 703 . . . . 5  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
1615adantrr 699 . . . 4  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
r  e. HAtoms  /\  r  C_  A ) )  -> 
( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH ) )
1716adantrr 699 . . 3  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH ) )
18 simpll 732 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  e.  CH )
199ad2antrl 710 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
( _|_ `  r
)  e.  CH )
20 chirred.1 . . . . . . . . 9  |-  A  e. 
CH
21 chsscon3 23003 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
227, 20, 21sylancl 645 . . . . . . . 8  |-  ( r  e. HAtoms  ->  ( r  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  r ) ) )
2322biimpa 472 . . . . . . 7  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
24 sstr 3357 . . . . . . 7  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  r ) )  ->  q  C_  ( _|_ `  r ) )
2523, 24sylan2 462 . . . . . 6  |-  ( ( q  C_  ( _|_ `  A )  /\  (
r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
2625adantll 696 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
27 lecm 23120 . . . . 5  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH  /\  q  C_  ( _|_ `  r
) )  ->  q  C_H  ( _|_ `  r
) )
2818, 19, 26, 27syl3anc 1185 . . . 4  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_H  ( _|_ `  r ) )
2928ad2ant2lr 730 . . 3  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
q  C_H  ( _|_ `  r ) )
30 chsscon3 23003 . . . . . . . . . . . . . 14  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
3120, 30mpan2 654 . . . . . . . . . . . . 13  |-  ( p  e.  CH  ->  (
p  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  p ) ) )
3231biimpa 472 . . . . . . . . . . . 12  |-  ( ( p  e.  CH  /\  p  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  p ) )
33 sstr 3357 . . . . . . . . . . . 12  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
3432, 33sylan2 462 . . . . . . . . . . 11  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e.  CH  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3534an12s 778 . . . . . . . . . 10  |-  ( ( p  e.  CH  /\  ( q  C_  ( _|_ `  A )  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3635ancom2s 779 . . . . . . . . 9  |-  ( ( p  e.  CH  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_  ( _|_ `  p ) )
3736adantll 696 . . . . . . . 8  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  q  C_  ( _|_ `  p ) )
38 choccl 22809 . . . . . . . . . . . 12  |-  ( p  e.  CH  ->  ( _|_ `  p )  e. 
CH )
39 lecm 23120 . . . . . . . . . . . 12  |-  ( ( q  e.  CH  /\  ( _|_ `  p )  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
4038, 39syl3an2 1219 . . . . . . . . . . 11  |-  ( ( q  e.  CH  /\  p  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
41403expia 1156 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  ( _|_ `  p ) ) )
42 cmcm2 23119 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_H  p  <->  q  C_H  ( _|_ `  p
) ) )
4341, 42sylibrd 227 . . . . . . . . 9  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  p )
)
4443adantr 453 . . . . . . . 8  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  ( q  C_  ( _|_ `  p )  ->  q  C_H  p
) )
4537, 44mpd 15 . . . . . . 7  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  q  C_H  p
)
461, 45sylanl2 634 . . . . . 6  |-  ( ( ( q  e.  CH  /\  p  e. HAtoms )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4746ancom1s 782 . . . . 5  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4847an4s 801 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4948adantr 453 . . 3  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
q  C_H  p )
50 fh2 23122 . . 3  |-  ( ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH )  /\  (
q  C_H  ( _|_ `  r )  /\  q  C_H  p ) )  -> 
( ( _|_ `  r
)  i^i  ( q  vH  p ) )  =  ( ( ( _|_ `  r )  i^i  q
)  vH  ( ( _|_ `  r )  i^i  p ) ) )
5117, 29, 49, 50syl12anc 1183 . 2  |-  ( ( ( ( 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  ( q  vH  p ) )  =  ( ( ( _|_ `  r )  i^i  q
)  vH  ( ( _|_ `  r )  i^i  p ) ) )
52 sseqin2 3561 . . . . . 6  |-  ( q 
C_  ( _|_ `  r
)  <->  ( ( _|_ `  r )  i^i  q
)  =  q )
5326, 52sylib 190 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
( ( _|_ `  r
)  i^i  q )  =  q )
5453ad2ant2lr 730 . . . 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
)  i^i  q )  =  q )
55 incom 3534 . . . . 5  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
5620chirredlem1 23894 . . . . . 6  |-  ( ( ( p  e. HAtoms  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  /\  ( ( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  (
p  vH  q )
) )  ->  (
p  i^i  ( _|_ `  r ) )  =  0H )
5756adantllr 701 . . . . 5  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( p  i^i  ( _|_ `  r ) )  =  0H )
5855, 57syl5eq 2481 . . . 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
)  i^i  p )  =  0H )
5954, 58oveq12d 6100 . . 3  |-  ( ( ( ( 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  q
)  vH  ( ( _|_ `  r )  i^i  p ) )  =  ( q  vH  0H ) )
60 chj0 23000 . . . . 5  |-  ( q  e.  CH  ->  (
q  vH  0H )  =  q )
6160adantr 453 . . . 4  |-  ( ( q  e.  CH  /\  q  C_  ( _|_ `  A
) )  ->  (
q  vH  0H )  =  q )
6261ad2antlr 709 . . 3  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  ( q  e. 
CH  /\  q  C_  ( _|_ `  A ) ) )  /\  (
( r  e. HAtoms  /\  r  C_  A )  /\  r  C_  ( p  vH  q
) ) )  -> 
( q  vH  0H )  =  q )
6359, 62eqtrd 2469 . 2  |-  ( ( ( ( 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  q
)  vH  ( ( _|_ `  r )  i^i  p ) )  =  q )
646, 51, 633eqtrd 2473 1  |-  ( ( ( ( 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 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320    C_ wss 3321   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CHcch 22433   _|_cort 22434    vH chj 22437   0Hc0h 22439    C_H ccm 22440  HAtomscat 22469
This theorem is referenced by:  chirredlem3  23896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cc 8316  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071  ax-hilex 22503  ax-hfvadd 22504  ax-hvcom 22505  ax-hvass 22506  ax-hv0cl 22507  ax-hvaddid 22508  ax-hfvmul 22509  ax-hvmulid 22510  ax-hvmulass 22511  ax-hvdistr1 22512  ax-hvdistr2 22513  ax-hvmul0 22514  ax-hfi 22582  ax-his1 22585  ax-his2 22586  ax-his3 22587  ax-his4 22588  ax-hcompl 22705
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-omul 6730  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-acn 7830  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-sum 12481  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-fbas 16700  df-fg 16701  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-ntr 17085  df-cls 17086  df-nei 17163  df-cn 17292  df-cnp 17293  df-lm 17294  df-haus 17380  df-tx 17595  df-hmeo 17788  df-fil 17879  df-fm 17971  df-flim 17972  df-flf 17973  df-xms 18351  df-ms 18352  df-tms 18353  df-cfil 19209  df-cau 19210  df-cmet 19211  df-grpo 21780  df-gid 21781  df-ginv 21782  df-gdiv 21783  df-ablo 21871  df-subgo 21891  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-vs 22079  df-nmcv 22080  df-ims 22081  df-dip 22198  df-ssp 22222  df-ph 22315  df-cbn 22366  df-hnorm 22472  df-hba 22473  df-hvsub 22475  df-hlim 22476  df-hcau 22477  df-sh 22710  df-ch 22725  df-oc 22755  df-ch0 22756  df-shs 22811  df-span 22812  df-chj 22813  df-chsup 22814  df-pjh 22898  df-cm 23086  df-cv 23783  df-at 23842
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