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Theorem chirredlem1 22931
Description: Lemma for chirredi 22935. (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 22885 . . . . . . 7  |-  ( r  e. HAtoms  ->  r  e.  CH )
2 chirred.1 . . . . . . . . 9  |-  A  e. 
CH
3 chsscon3 22040 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
42, 3mpan2 655 . . . . . . . 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 3161 . . . . . 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 22885 . . . . . . . . 9  |-  ( p  e. HAtoms  ->  p  e.  CH )
10 atne0 22886 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  r  =/=  0H )
1110neneqd 2437 . . . . . . . . . . . 12  |-  ( r  e. HAtoms  ->  -.  r  =  0H )
1211ad3antrrr 713 . . . . . . . . . . 11  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  r  =  0H )
13 simplr 734 . . . . . . . . . . . . . . 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 21846 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
151, 14syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
16 chlej1 22050 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p  e.  CH  /\  ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  p  C_  ( _|_ `  r ) )  -> 
( p  vH  q
)  C_  ( ( _|_ `  r )  vH  q ) )
17163exp1 1172 . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  (
p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2019adantllr 702 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2120adantlr 698 . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . 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 22053 . . . . . . . . . . . . . . . . . . . 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 634 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r
) )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2726an32s 782 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  q  e.  CH )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2827adantrl 699 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) )
2928ad2antrr 709 . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . 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 22036 . . . . . . . . . . . . . . 15  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  <->  r  =  0H ) )
3332biimpd 200 . . . . . . . . . . . . . 14  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  ->  r  =  0H ) )
341, 33syl 17 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  ( r  C_  ( _|_ `  r )  ->  r  =  0H ) )
3534ad3antrrr 713 . . . . . . . . . . . 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 425 . . . . . . . . 9  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( r  C_  ( p  vH  q
)  ->  -.  p  C_  ( _|_ `  r
) ) )
399, 38sylanr1 636 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  -.  p  C_  ( _|_ `  r ) ) )
40 atnssm0 22917 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  i^i  p )  =  0H ) )
41 incom 3336 . . . . . . . . . . . 12  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
4241eqeq1i 2265 . . . . . . . . . . 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 730 . . . . . . . 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 598 . . . . . 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 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 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 581 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 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    i^i cin 3126    C_ wss 3127   ` cfv 4673  (class class class)co 5792   CHcch 21470   _|_cort 21471    vH chj 21474   0Hc0h 21476  HAtomscat 21506
This theorem is referenced by:  chirredlem2  22932
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cc 8029  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785  ax-hilex 21540  ax-hfvadd 21541  ax-hvcom 21542  ax-hvass 21543  ax-hv0cl 21544  ax-hvaddid 21545  ax-hfvmul 21546  ax-hvmulid 21547  ax-hvmulass 21548  ax-hvdistr1 21549  ax-hvdistr2 21550  ax-hvmul0 21551  ax-hfi 21619  ax-his1 21622  ax-his2 21623  ax-his3 21624  ax-his4 21625  ax-hcompl 21742
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-omul 6452  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-acn 7543  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-ioo 10627  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-fl 10892  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-rlim 11929  df-sum 12125  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-rest 13290  df-topn 13291  df-topgen 13307  df-pt 13308  df-prds 13311  df-xrs 13366  df-0g 13367  df-gsum 13368  df-qtop 13373  df-imas 13374  df-xps 13376  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-submnd 14379  df-mulg 14455  df-cntz 14756  df-cmn 15054  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cld 16719  df-ntr 16720  df-cls 16721  df-nei 16798  df-cn 16920  df-cnp 16921  df-lm 16922  df-haus 17006  df-tx 17220  df-hmeo 17409  df-fbas 17483  df-fg 17484  df-fil 17504  df-fm 17596  df-flim 17597  df-flf 17598  df-xms 17848  df-ms 17849  df-tms 17850  df-cfil 18644  df-cau 18645  df-cmet 18646  df-grpo 20819  df-gid 20820  df-ginv 20821  df-gdiv 20822  df-ablo 20910  df-subgo 20930  df-vc 21063  df-nv 21109  df-va 21112  df-ba 21113  df-sm 21114  df-0v 21115  df-vs 21116  df-nmcv 21117  df-ims 21118  df-dip 21235  df-ssp 21259  df-ph 21352  df-cbn 21403  df-hnorm 21509  df-hba 21510  df-hvsub 21512  df-hlim 21513  df-hcau 21514  df-sh 21747  df-ch 21762  df-oc 21792  df-ch0 21793  df-shs 21848  df-span 21849  df-chj 21850  df-chsup 21851  df-pjh 21935  df-cv 22820  df-at 22879
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