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Theorem chirredlem2 27002
Description: Lemma for chirredi 27005. (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 26955 . . . . . 6  |-  ( p  e. HAtoms  ->  p  e.  CH )
2 chjcom 26116 . . . . . 6  |-  ( ( p  e.  CH  /\  q  e.  CH )  ->  ( p  vH  q
)  =  ( q  vH  p ) )
31, 2sylan 471 . . . . 5  |-  ( ( p  e. HAtoms  /\  q  e.  CH )  ->  (
p  vH  q )  =  ( q  vH  p ) )
43ad2ant2r 746 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
( p  vH  q
)  =  ( q  vH  p ) )
54adantr 465 . . 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 3700 . 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 26955 . . . . . . . . . . 11  |-  ( r  e. HAtoms  ->  r  e.  CH )
8 choccl 25916 . . . . . . . . . . 11  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
97, 8syl 16 . . . . . . . . . 10  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
10 id 22 . . . . . . . . . 10  |-  ( q  e.  CH  ->  q  e.  CH )
119, 10, 13anim123i 1181 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  q  e.  CH  /\  p  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
12113com13 1201 . . . . . . . 8  |-  ( ( p  e. HAtoms  /\  q  e.  CH  /\  r  e. HAtoms
)  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
13123expa 1196 . . . . . . 7  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r )  e. 
CH  /\  q  e.  CH 
/\  p  e.  CH ) )
1413adantllr 718 . . . . . 6  |-  ( ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  q  e.  CH )  /\  r  e. HAtoms )  ->  ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH  /\  p  e.  CH ) )
1514adantlrr 720 . . . . 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 716 . . . 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 716 . . 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 753 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  e.  CH )
199ad2antrl 727 . . . . 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 26110 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
227, 20, 21sylancl 662 . . . . . . . 8  |-  ( r  e. HAtoms  ->  ( r  C_  A 
<->  ( _|_ `  A
)  C_  ( _|_ `  r ) ) )
2322biimpa 484 . . . . . . 7  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
24 sstr 3512 . . . . . . 7  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  r ) )  ->  q  C_  ( _|_ `  r ) )
2523, 24sylan2 474 . . . . . 6  |-  ( ( q  C_  ( _|_ `  A )  /\  (
r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
2625adantll 713 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_  ( _|_ `  r ) )
27 lecm 26227 . . . . 5  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH  /\  q  C_  ( _|_ `  r
) )  ->  q  C_H  ( _|_ `  r
) )
2818, 19, 26, 27syl3anc 1228 . . . 4  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
q  C_H  ( _|_ `  r ) )
2928ad2ant2lr 747 . . 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 26110 . . . . . . . . . . . . . 14  |-  ( ( p  e.  CH  /\  A  e.  CH )  ->  ( p  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  p
) ) )
3120, 30mpan2 671 . . . . . . . . . . . . 13  |-  ( p  e.  CH  ->  (
p  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  p ) ) )
3231biimpa 484 . . . . . . . . . . . 12  |-  ( ( p  e.  CH  /\  p  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  p ) )
33 sstr 3512 . . . . . . . . . . . 12  |-  ( ( q  C_  ( _|_ `  A )  /\  ( _|_ `  A )  C_  ( _|_ `  p ) )  ->  q  C_  ( _|_ `  p ) )
3432, 33sylan2 474 . . . . . . . . . . 11  |-  ( ( q  C_  ( _|_ `  A )  /\  (
p  e.  CH  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3534an12s 799 . . . . . . . . . 10  |-  ( ( p  e.  CH  /\  ( q  C_  ( _|_ `  A )  /\  p  C_  A ) )  ->  q  C_  ( _|_ `  p ) )
3635ancom2s 800 . . . . . . . . 9  |-  ( ( p  e.  CH  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_  ( _|_ `  p ) )
3736adantll 713 . . . . . . . 8  |-  ( ( ( q  e.  CH  /\  p  e.  CH )  /\  ( p  C_  A  /\  q  C_  ( _|_ `  A ) ) )  ->  q  C_  ( _|_ `  p ) )
38 choccl 25916 . . . . . . . . . . . 12  |-  ( p  e.  CH  ->  ( _|_ `  p )  e. 
CH )
39 lecm 26227 . . . . . . . . . . . 12  |-  ( ( q  e.  CH  /\  ( _|_ `  p )  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
4038, 39syl3an2 1262 . . . . . . . . . . 11  |-  ( ( q  e.  CH  /\  p  e.  CH  /\  q  C_  ( _|_ `  p
) )  ->  q  C_H  ( _|_ `  p
) )
41403expia 1198 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  ( _|_ `  p ) ) )
42 cmcm2 26226 . . . . . . . . . 10  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_H  p  <->  q  C_H  ( _|_ `  p
) ) )
4341, 42sylibrd 234 . . . . . . . . 9  |-  ( ( q  e.  CH  /\  p  e.  CH )  ->  ( q  C_  ( _|_ `  p )  -> 
q  C_H  p )
)
4443adantr 465 . . . . . . . 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 651 . . . . . 6  |-  ( ( ( q  e.  CH  /\  p  e. HAtoms )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4746ancom1s 803 . . . . 5  |-  ( ( ( p  e. HAtoms  /\  q  e.  CH )  /\  (
p  C_  A  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4847an4s 824 . . . 4  |-  ( ( ( p  e. HAtoms  /\  p  C_  A )  /\  (
q  e.  CH  /\  q  C_  ( _|_ `  A
) ) )  -> 
q  C_H  p )
4948adantr 465 . . 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 26229 . . 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 1226 . 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 3717 . . . . . 6  |-  ( q 
C_  ( _|_ `  r
)  <->  ( ( _|_ `  r )  i^i  q
)  =  q )
5326, 52sylib 196 . . . . 5  |-  ( ( ( q  e.  CH  /\  q  C_  ( _|_ `  A ) )  /\  ( r  e. HAtoms  /\  r  C_  A ) )  -> 
( ( _|_ `  r
)  i^i  q )  =  q )
5453ad2ant2lr 747 . . . 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 3691 . . . . 5  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
5620chirredlem1 27001 . . . . . 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 718 . . . . 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 2520 . . . 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 6301 . . 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 26107 . . . . 5  |-  ( q  e.  CH  ->  (
q  vH  0H )  =  q )
6160adantr 465 . . . 4  |-  ( ( q  e.  CH  /\  q  C_  ( _|_ `  A
) )  ->  (
q  vH  0H )  =  q )
6261ad2antlr 726 . . 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 2508 . 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 2512 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   CHcch 25538   _|_cort 25539    vH chj 25542   0Hc0h 25544    C_H ccm 25545  HAtomscat 25574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cc 8814  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571  ax-hilex 25608  ax-hfvadd 25609  ax-hvcom 25610  ax-hvass 25611  ax-hv0cl 25612  ax-hvaddid 25613  ax-hfvmul 25614  ax-hvmulid 25615  ax-hvmulass 25616  ax-hvdistr1 25617  ax-hvdistr2 25618  ax-hvmul0 25619  ax-hfi 25688  ax-his1 25691  ax-his2 25692  ax-his3 25693  ax-his4 25694  ax-hcompl 25811
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-acn 8322  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-cn 19510  df-cnp 19511  df-lm 19512  df-haus 19598  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cfil 21445  df-cau 21446  df-cmet 21447  df-grpo 24885  df-gid 24886  df-ginv 24887  df-gdiv 24888  df-ablo 24976  df-subgo 24996  df-vc 25131  df-nv 25177  df-va 25180  df-ba 25181  df-sm 25182  df-0v 25183  df-vs 25184  df-nmcv 25185  df-ims 25186  df-dip 25303  df-ssp 25327  df-ph 25420  df-cbn 25471  df-hnorm 25577  df-hba 25578  df-hvsub 25580  df-hlim 25581  df-hcau 25582  df-sh 25816  df-ch 25831  df-oc 25862  df-ch0 25863  df-shs 25918  df-span 25919  df-chj 25920  df-chsup 25921  df-pjh 26005  df-cm 26193  df-cv 26890  df-at 26949
This theorem is referenced by:  chirredlem3  27003
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