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Theorem chirredlem1 23898
Description: Lemma for chirredi 23902. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
chirred.1  |-  A  e. 
CH
Assertion
Ref Expression
chirredlem1  |-  ( ( ( 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 )
Distinct variable group:    q, p, r, A

Proof of Theorem chirredlem1
StepHypRef Expression
1 atelch 23852 . . . . . . 7  |-  ( r  e. HAtoms  ->  r  e.  CH )
2 chirred.1 . . . . . . . . 9  |-  A  e. 
CH
3 chsscon3 23007 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
42, 3mpan2 654 . . . . . . . 8  |-  ( r  e.  CH  ->  (
r  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  r ) ) )
54biimpa 472 . . . . . . 7  |-  ( ( r  e.  CH  /\  r  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  r ) )
61, 5sylan 459 . . . . . 6  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
7 sstr2 3357 . . . . . 6  |-  ( q 
C_  ( _|_ `  A
)  ->  ( ( _|_ `  A )  C_  ( _|_ `  r )  ->  q  C_  ( _|_ `  r ) ) )
86, 7syl5 31 . . . . 5  |-  ( q 
C_  ( _|_ `  A
)  ->  ( (
r  e. HAtoms  /\  r  C_  A )  ->  q  C_  ( _|_ `  r
) ) )
9 atelch 23852 . . . . . . . . 9  |-  ( p  e. HAtoms  ->  p  e.  CH )
10 atne0 23853 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  r  =/=  0H )
1110neneqd 2619 . . . . . . . . . . . 12  |-  ( r  e. HAtoms  ->  -.  r  =  0H )
1211ad3antrrr 712 . . . . . . . . . . 11  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  r  =  0H )
13 simplr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e. 
CH  /\  q  e.  CH ) )  /\  r  C_  ( p  vH  q
) )  /\  p  C_  ( _|_ `  r
) )  ->  r  C_  ( p  vH  q
) )
14 choccl 22813 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
151, 14syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
16 chlej1 23017 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p  e.  CH  /\  ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  p  C_  ( _|_ `  r ) )  -> 
( p  vH  q
)  C_  ( ( _|_ `  r )  vH  q ) )
17163exp1 1170 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  CH  ->  (
( _|_ `  r
)  e.  CH  ->  ( q  e.  CH  ->  ( p  C_  ( _|_ `  r )  ->  (
p  vH  q )  C_  ( ( _|_ `  r
)  vH  q )
) ) ) )
1815, 17syl5com 29 . . . . . . . . . . . . . . . . . 18  |-  ( r  e. HAtoms  ->  ( p  e. 
CH  ->  ( q  e. 
CH  ->  ( p  C_  ( _|_ `  r )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
) ) ) )
1918imp42 579 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  (
p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2019adantllr 701 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2120adantlr 697 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e. 
CH  /\  q  e.  CH ) )  /\  r  C_  ( p  vH  q
) )  /\  p  C_  ( _|_ `  r
) )  ->  (
p  vH  q )  C_  ( ( _|_ `  r
)  vH  q )
)
2213, 21sstrd 3360 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e. 
CH  /\  q  e.  CH ) )  /\  r  C_  ( p  vH  q
) )  /\  p  C_  ( _|_ `  r
) )  ->  r  C_  ( ( _|_ `  r
)  vH  q )
)
23 chlejb2 23020 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH )  -> 
( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2423ancoms 441 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  ->  ( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2524biimpa 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r ) )  -> 
( ( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2615, 25sylanl1 633 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r
) )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2726an32s 781 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  q  e.  CH )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2827adantrl 698 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) )
2928ad2antrr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e. 
CH  /\  q  e.  CH ) )  /\  r  C_  ( p  vH  q
) )  /\  p  C_  ( _|_ `  r
) )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
3022, 29sseqtrd 3386 . . . . . . . . . . . . 13  |-  ( ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e. 
CH  /\  q  e.  CH ) )  /\  r  C_  ( p  vH  q
) )  /\  p  C_  ( _|_ `  r
) )  ->  r  C_  ( _|_ `  r
) )
3130ex 425 . . . . . . . . . . . 12  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  (
p  C_  ( _|_ `  r )  ->  r  C_  ( _|_ `  r
) ) )
32 chssoc 23003 . . . . . . . . . . . . . . 15  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  <->  r  =  0H ) )
3332biimpd 200 . . . . . . . . . . . . . 14  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  ->  r  =  0H ) )
341, 33syl 16 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  ( r  C_  ( _|_ `  r )  ->  r  =  0H ) )
3534ad3antrrr 712 . . . . . . . . . . . 12  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  (
r  C_  ( _|_ `  r )  ->  r  =  0H ) )
3631, 35syld 43 . . . . . . . . . . 11  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  (
p  C_  ( _|_ `  r )  ->  r  =  0H ) )
3712, 36mtod 171 . . . . . . . . . 10  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  p  C_  ( _|_ `  r
) )
3837ex 425 . . . . . . . . 9  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( r  C_  ( p  vH  q
)  ->  -.  p  C_  ( _|_ `  r
) ) )
399, 38sylanr1 635 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  -.  p  C_  ( _|_ `  r ) ) )
40 atnssm0 23884 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  i^i  p )  =  0H ) )
41 incom 3535 . . . . . . . . . . . 12  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
4241eqeq1i 2445 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  i^i  p )  =  0H  <->  ( p  i^i  ( _|_ `  r
) )  =  0H )
4340, 42syl6bb 254 . . . . . . . . . 10  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4415, 43sylan 459 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r
)  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4544ad2ant2r 729 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( -.  p  C_  ( _|_ `  r )  <-> 
( p  i^i  ( _|_ `  r ) )  =  0H ) )
4639, 45sylibd 207 . . . . . . 7  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) )
4746exp43 597 . . . . . 6  |-  ( r  e. HAtoms  ->  ( q  C_  ( _|_ `  r )  ->  ( p  e. HAtoms  ->  ( q  e.  CH  ->  ( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) ) ) ) )
4847adantr 453 . . . . 5  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  (
q  C_  ( _|_ `  r )  ->  (
p  e. HAtoms  ->  ( q  e.  CH  ->  (
r  C_  ( p  vH  q )  ->  (
p  i^i  ( _|_ `  r ) )  =  0H ) ) ) ) )
498, 48sylcom 28 . . . 4  |-  ( q 
C_  ( _|_ `  A
)  ->  ( (
r  e. HAtoms  /\  r  C_  A )  ->  (
p  e. HAtoms  ->  ( q  e.  CH  ->  (
r  C_  ( p  vH  q )  ->  (
p  i^i  ( _|_ `  r ) )  =  0H ) ) ) ) )
5049com4t 82 . . 3  |-  ( 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 ) ) ) ) )
5150imp3a 422 . 2  |-  ( 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 ) ) ) )
5251imp43 580 1  |-  ( ( ( 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 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   ` cfv 5457  (class class class)co 6084   CHcch 22437   _|_cort 22438    vH chj 22441   0Hc0h 22443  HAtomscat 22473
This theorem is referenced by:  chirredlem2  23899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cc 8320  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075  ax-hilex 22507  ax-hfvadd 22508  ax-hvcom 22509  ax-hvass 22510  ax-hv0cl 22511  ax-hvaddid 22512  ax-hfvmul 22513  ax-hvmulid 22514  ax-hvmulass 22515  ax-hvdistr1 22516  ax-hvdistr2 22517  ax-hvmul0 22518  ax-hfi 22586  ax-his1 22589  ax-his2 22590  ax-his3 22591  ax-his4 22592  ax-hcompl 22709
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-cn 17296  df-cnp 17297  df-lm 17298  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cfil 19213  df-cau 19214  df-cmet 19215  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-subgo 21895  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-vs 22083  df-nmcv 22084  df-ims 22085  df-dip 22202  df-ssp 22226  df-ph 22319  df-cbn 22370  df-hnorm 22476  df-hba 22477  df-hvsub 22479  df-hlim 22480  df-hcau 22481  df-sh 22714  df-ch 22729  df-oc 22759  df-ch0 22760  df-shs 22815  df-span 22816  df-chj 22817  df-chsup 22818  df-pjh 22902  df-cv 23787  df-at 23846
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