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Theorem chirredlem1 23854
Description: Lemma for chirredi 23858. (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 23808 . . . . . . 7  |-  ( r  e. HAtoms  ->  r  e.  CH )
2 chirred.1 . . . . . . . . 9  |-  A  e. 
CH
3 chsscon3 22963 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
42, 3mpan2 653 . . . . . . . 8  |-  ( r  e.  CH  ->  (
r  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  r ) ) )
54biimpa 471 . . . . . . 7  |-  ( ( r  e.  CH  /\  r  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  r ) )
61, 5sylan 458 . . . . . 6  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
7 sstr2 3323 . . . . . 6  |-  ( q 
C_  ( _|_ `  A
)  ->  ( ( _|_ `  A )  C_  ( _|_ `  r )  ->  q  C_  ( _|_ `  r ) ) )
86, 7syl5 30 . . . . 5  |-  ( q 
C_  ( _|_ `  A
)  ->  ( (
r  e. HAtoms  /\  r  C_  A )  ->  q  C_  ( _|_ `  r
) ) )
9 atelch 23808 . . . . . . . . 9  |-  ( p  e. HAtoms  ->  p  e.  CH )
10 atne0 23809 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  r  =/=  0H )
1110neneqd 2591 . . . . . . . . . . . 12  |-  ( r  e. HAtoms  ->  -.  r  =  0H )
1211ad3antrrr 711 . . . . . . . . . . 11  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  r  =  0H )
13 simplr 732 . . . . . . . . . . . . . . 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 22769 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
151, 14syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
16 chlej1 22973 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p  e.  CH  /\  ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  p  C_  ( _|_ `  r ) )  -> 
( p  vH  q
)  C_  ( ( _|_ `  r )  vH  q ) )
17163exp1 1169 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  CH  ->  (
( _|_ `  r
)  e.  CH  ->  ( q  e.  CH  ->  ( p  C_  ( _|_ `  r )  ->  (
p  vH  q )  C_  ( ( _|_ `  r
)  vH  q )
) ) ) )
1815, 17syl5com 28 . . . . . . . . . . . . . . . . . 18  |-  ( r  e. HAtoms  ->  ( p  e. 
CH  ->  ( q  e. 
CH  ->  ( p  C_  ( _|_ `  r )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
) ) ) )
1918imp42 578 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  (
p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2019adantllr 700 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2120adantlr 696 . . . . . . . . . . . . . . 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 3326 . . . . . . . . . . . . . 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 22976 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH )  -> 
( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2423ancoms 440 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  ->  ( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2524biimpa 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r ) )  -> 
( ( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2615, 25sylanl1 632 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r
) )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2726an32s 780 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  q  e.  CH )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2827adantrl 697 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) )
2928ad2antrr 707 . . . . . . . . . . . . . 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 3352 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 22959 . . . . . . . . . . . . . . 15  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  <->  r  =  0H ) )
3332biimpd 199 . . . . . . . . . . . . . 14  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  ->  r  =  0H ) )
341, 33syl 16 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  ( r  C_  ( _|_ `  r )  ->  r  =  0H ) )
3534ad3antrrr 711 . . . . . . . . . . . 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 42 . . . . . . . . . . 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 170 . . . . . . . . . 10  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  p  C_  ( _|_ `  r
) )
3837ex 424 . . . . . . . . 9  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( r  C_  ( p  vH  q
)  ->  -.  p  C_  ( _|_ `  r
) ) )
399, 38sylanr1 634 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  -.  p  C_  ( _|_ `  r ) ) )
40 atnssm0 23840 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  i^i  p )  =  0H ) )
41 incom 3501 . . . . . . . . . . . 12  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
4241eqeq1i 2419 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  i^i  p )  =  0H  <->  ( p  i^i  ( _|_ `  r
) )  =  0H )
4340, 42syl6bb 253 . . . . . . . . . 10  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4415, 43sylan 458 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r
)  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4544ad2ant2r 728 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( -.  p  C_  ( _|_ `  r )  <-> 
( p  i^i  ( _|_ `  r ) )  =  0H ) )
4639, 45sylibd 206 . . . . . . 7  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) )
4746exp43 596 . . . . . 6  |-  ( r  e. HAtoms  ->  ( q  C_  ( _|_ `  r )  ->  ( p  e. HAtoms  ->  ( q  e.  CH  ->  ( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) ) ) ) )
4847adantr 452 . . . . 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 27 . . . 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 81 . . 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 421 . 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 579 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3287    C_ wss 3288   ` cfv 5421  (class class class)co 6048   CHcch 22393   _|_cort 22394    vH chj 22397   0Hc0h 22399  HAtomscat 22429
This theorem is referenced by:  chirredlem2  23855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cc 8279  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034  ax-hilex 22463  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvmulass 22471  ax-hvdistr1 22472  ax-hvdistr2 22473  ax-hvmul0 22474  ax-hfi 22542  ax-his1 22545  ax-his2 22546  ax-his3 22547  ax-his4 22548  ax-hcompl 22665
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-acn 7793  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-cn 17253  df-cnp 17254  df-lm 17255  df-haus 17341  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cfil 19169  df-cau 19170  df-cmet 19171  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-subgo 21851  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040  df-ims 22041  df-dip 22158  df-ssp 22182  df-ph 22275  df-cbn 22326  df-hnorm 22432  df-hba 22433  df-hvsub 22435  df-hlim 22436  df-hcau 22437  df-sh 22670  df-ch 22685  df-oc 22715  df-ch0 22716  df-shs 22771  df-span 22772  df-chj 22773  df-chsup 22774  df-pjh 22858  df-cv 23743  df-at 23802
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