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Theorem chirredlem2 22987
Description: Lemma for chirredi 22990. (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 22940 . . . . . 6  |-  ( p  e. HAtoms  ->  p  e.  CH )
2 chjcom 22101 . . . . . 6  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( p  vH  q
)  =  ( q  vH  p ) )
31, 2sylan 457 . . . . 5  |-  ( ( p  e. HAtoms  /\  q  e.  CH )  ->  (
p  vH  q )  =  ( q  vH  p ) )
43ad2ant2r 727 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( p  vH  q
)  =  ( q  vH  p ) )
54adantr 451 . . 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 3383 . 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 22940 . . . . . . . . . . 11  |-  ( r  e. HAtoms  ->  r  e.  CH )
8 choccl 21901 . . . . . . . . . . 11  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
97, 8syl 15 . . . . . . . . . 10  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
10 id 19 . . . . . . . . . 10  |-  ( q  e.  CH  ->  q  e.  CH )
119, 10, 13anim123i 1137 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  q  e.  CH  /\  p  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
12113com13 1156 . . . . . . . 8  |-  ( ( p  e. HAtoms  /\  q  e.  CH  /\  r  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
13123expa 1151 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
1413adantllr 699 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH ) )
1514adantlrr 701 . . . . 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 697 . . . 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 697 . . 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 730 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  e.  CH )
199ad2antrl 708 . . . . 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 22095 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
227, 20, 21sylancl 643 . . . . . . . 8  |-  ( r  e. HAtoms  ->  ( r  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  r ) ) )
2322biimpa 470 . . . . . . 7  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
24 sstr 3200 . . . . . . 7  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  r ) )  ->  q  C_  ( _|_ `  r ) )
2523, 24sylan2 460 . . . . . 6  |-  ( ( q  C_  ( _|_ `  A )  /\  (
r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
2625adantll 694 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
27 lecm 22212 . . . . 5  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH  /\  q  C_  ( _|_ `  r
) )  ->  q  C_H  ( _|_ `  r
) )
2818, 19, 26, 27syl3anc 1182 . . . 4  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_H  ( _|_ `  r ) )
2928ad2ant2lr 728 . . 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 22095 . . . . . . . . . . . . . 14  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
3120, 30mpan2 652 . . . . . . . . . . . . 13  |-  ( p  e.  CH  ->  (
p  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  p ) ) )
3231biimpa 470 . . . . . . . . . . . 12  |-  ( ( p  e.  CH  /\  p  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  p ) )
33 sstr 3200 . . . . . . . . . . . 12  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
3432, 33sylan2 460 . . . . . . . . . . 11  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e.  CH  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3534an12s 776 . . . . . . . . . 10  |-  ( ( p  e.  CH  /\  ( q  C_  ( _|_ `  A )  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3635ancom2s 777 . . . . . . . . 9  |-  ( ( p  e.  CH  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_  ( _|_ `  p ) )
3736adantll 694 . . . . . . . 8  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  q  C_  ( _|_ `  p ) )
38 choccl 21901 . . . . . . . . . . . 12  |-  ( p  e.  CH  ->  ( _|_ `  p )  e. 
CH )
39 lecm 22212 . . . . . . . . . . . 12  |-  ( ( q  e.  CH  /\  ( _|_ `  p )  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
4038, 39syl3an2 1216 . . . . . . . . . . 11  |-  ( ( q  e.  CH  /\  p  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
41403expia 1153 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  ( _|_ `  p ) ) )
42 cmcm2 22211 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_H  p  <->  q  C_H  ( _|_ `  p
) ) )
4341, 42sylibrd 225 . . . . . . . . 9  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  p )
)
4443adantr 451 . . . . . . . 8  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  ( q  C_  ( _|_ `  p )  ->  q  C_H  p
) )
4537, 44mpd 14 . . . . . . 7  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  q  C_H  p
)
461, 45sylanl2 632 . . . . . 6  |-  ( ( ( q  e.  CH  /\  p  e. HAtoms )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4746ancom1s 780 . . . . 5  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4847an4s 799 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4948adantr 451 . . 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 22214 . . 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 1180 . 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 3401 . . . . . 6  |-  ( q 
C_  ( _|_ `  r
)  <->  ( ( _|_ `  r )  i^i  q
)  =  q )
5326, 52sylib 188 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
( ( _|_ `  r
)  i^i  q )  =  q )
5453ad2ant2lr 728 . . . 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 3374 . . . . 5  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
5620chirredlem1 22986 . . . . . 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 699 . . . . 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 2340 . . . 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 5892 . . 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 22092 . . . . 5  |-  ( q  e.  CH  ->  (
q  vH  0H )  =  q )
6160adantr 451 . . . 4  |-  ( ( q  e.  CH  /\  q  C_  ( _|_ `  A
) )  ->  (
q  vH  0H )  =  q )
6261ad2antlr 707 . . 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 2328 . 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 2332 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CHcch 21525   _|_cort 21526    vH chj 21529   0Hc0h 21531    C_H ccm 21532  HAtomscat 21561
This theorem is referenced by:  chirredlem3  22988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680  ax-hcompl 21797
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-cn 16973  df-cnp 16974  df-lm 16975  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cfil 18697  df-cau 18698  df-cmet 18699  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-dip 21290  df-ssp 21314  df-ph 21407  df-cbn 21458  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-hcau 21569  df-sh 21802  df-ch 21817  df-oc 21847  df-ch0 21848  df-shs 21903  df-span 21904  df-chj 21905  df-chsup 21906  df-pjh 21990  df-cm 22178  df-cv 22875  df-at 22934
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