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Theorem chirredlem1 23078
Description: Lemma for chirredi 23082. (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 23032 . . . . . . 7  |-  ( r  e. HAtoms  ->  r  e.  CH )
2 chirred.1 . . . . . . . . 9  |-  A  e. 
CH
3 chsscon3 22187 . . . . . . . . 9  |-  ( ( r  e.  CH  /\  A  e.  CH )  ->  ( r  C_  A  <->  ( _|_ `  A ) 
C_  ( _|_ `  r
) ) )
42, 3mpan2 652 . . . . . . . 8  |-  ( r  e.  CH  ->  (
r  C_  A  <->  ( _|_ `  A )  C_  ( _|_ `  r ) ) )
54biimpa 470 . . . . . . 7  |-  ( ( r  e.  CH  /\  r  C_  A )  -> 
( _|_ `  A
)  C_  ( _|_ `  r ) )
61, 5sylan 457 . . . . . 6  |-  ( ( r  e. HAtoms  /\  r  C_  A )  ->  ( _|_ `  A )  C_  ( _|_ `  r ) )
7 sstr2 3262 . . . . . 6  |-  ( q 
C_  ( _|_ `  A
)  ->  ( ( _|_ `  A )  C_  ( _|_ `  r )  ->  q  C_  ( _|_ `  r ) ) )
86, 7syl5 28 . . . . 5  |-  ( q 
C_  ( _|_ `  A
)  ->  ( (
r  e. HAtoms  /\  r  C_  A )  ->  q  C_  ( _|_ `  r
) ) )
9 atelch 23032 . . . . . . . . 9  |-  ( p  e. HAtoms  ->  p  e.  CH )
10 atne0 23033 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  r  =/=  0H )
1110neneqd 2537 . . . . . . . . . . . 12  |-  ( r  e. HAtoms  ->  -.  r  =  0H )
1211ad3antrrr 710 . . . . . . . . . . 11  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  r  =  0H )
13 simplr 731 . . . . . . . . . . . . . . 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 21993 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  e.  CH  ->  ( _|_ `  r )  e. 
CH )
151, 14syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( r  e. HAtoms  ->  ( _|_ `  r
)  e.  CH )
16 chlej1 22197 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p  e.  CH  /\  ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  p  C_  ( _|_ `  r ) )  -> 
( p  vH  q
)  C_  ( ( _|_ `  r )  vH  q ) )
17163exp1 1167 . . . . . . . . . . . . . . . . . . 19  |-  ( p  e.  CH  ->  (
( _|_ `  r
)  e.  CH  ->  ( q  e.  CH  ->  ( p  C_  ( _|_ `  r )  ->  (
p  vH  q )  C_  ( ( _|_ `  r
)  vH  q )
) ) ) )
1815, 17syl5com 26 . . . . . . . . . . . . . . . . . 18  |-  ( r  e. HAtoms  ->  ( p  e. 
CH  ->  ( q  e. 
CH  ->  ( p  C_  ( _|_ `  r )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
) ) ) )
1918imp42 577 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  (
p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2019adantllr 699 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  p  C_  ( _|_ `  r ) )  ->  ( p  vH  q )  C_  (
( _|_ `  r
)  vH  q )
)
2120adantlr 695 . . . . . . . . . . . . . . 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 3265 . . . . . . . . . . . . . 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 22200 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( q  e.  CH  /\  ( _|_ `  r )  e.  CH )  -> 
( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2423ancoms 439 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  ->  ( q  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) ) )
2524biimpa 470 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( _|_ `  r
)  e.  CH  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r ) )  -> 
( ( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2615, 25sylanl1 631 . . . . . . . . . . . . . . . . 17  |-  ( ( ( r  e. HAtoms  /\  q  e.  CH )  /\  q  C_  ( _|_ `  r
) )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2726an32s 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  q  e.  CH )  ->  (
( _|_ `  r
)  vH  q )  =  ( _|_ `  r
) )
2827adantrl 696 . . . . . . . . . . . . . . 15  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( ( _|_ `  r )  vH  q )  =  ( _|_ `  r ) )
2928ad2antrr 706 . . . . . . . . . . . . . 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 3290 . . . . . . . . . . . . 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 423 . . . . . . . . . . . 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 22183 . . . . . . . . . . . . . . 15  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  <->  r  =  0H ) )
3332biimpd 198 . . . . . . . . . . . . . 14  |-  ( r  e.  CH  ->  (
r  C_  ( _|_ `  r )  ->  r  =  0H ) )
341, 33syl 15 . . . . . . . . . . . . 13  |-  ( r  e. HAtoms  ->  ( r  C_  ( _|_ `  r )  ->  r  =  0H ) )
3534ad3antrrr 710 . . . . . . . . . . . 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 40 . . . . . . . . . . 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 168 . . . . . . . . . 10  |-  ( ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r ) )  /\  ( p  e.  CH  /\  q  e.  CH )
)  /\  r  C_  ( p  vH  q
) )  ->  -.  p  C_  ( _|_ `  r
) )
3837ex 423 . . . . . . . . 9  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e.  CH  /\  q  e.  CH )
)  ->  ( r  C_  ( p  vH  q
)  ->  -.  p  C_  ( _|_ `  r
) ) )
399, 38sylanr1 633 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  -.  p  C_  ( _|_ `  r ) ) )
40 atnssm0 23064 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( ( _|_ `  r )  i^i  p )  =  0H ) )
41 incom 3437 . . . . . . . . . . . 12  |-  ( ( _|_ `  r )  i^i  p )  =  ( p  i^i  ( _|_ `  r ) )
4241eqeq1i 2365 . . . . . . . . . . 11  |-  ( ( ( _|_ `  r
)  i^i  p )  =  0H  <->  ( p  i^i  ( _|_ `  r
) )  =  0H )
4340, 42syl6bb 252 . . . . . . . . . 10  |-  ( ( ( _|_ `  r
)  e.  CH  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r )  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4415, 43sylan 457 . . . . . . . . 9  |-  ( ( r  e. HAtoms  /\  p  e. HAtoms )  ->  ( -.  p  C_  ( _|_ `  r
)  <->  ( p  i^i  ( _|_ `  r
) )  =  0H ) )
4544ad2ant2r 727 . . . . . . . 8  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( -.  p  C_  ( _|_ `  r )  <-> 
( p  i^i  ( _|_ `  r ) )  =  0H ) )
4639, 45sylibd 205 . . . . . . 7  |-  ( ( ( r  e. HAtoms  /\  q  C_  ( _|_ `  r
) )  /\  (
p  e. HAtoms  /\  q  e.  CH ) )  -> 
( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) )
4746exp43 595 . . . . . 6  |-  ( r  e. HAtoms  ->  ( q  C_  ( _|_ `  r )  ->  ( p  e. HAtoms  ->  ( q  e.  CH  ->  ( r  C_  (
p  vH  q )  ->  ( p  i^i  ( _|_ `  r ) )  =  0H ) ) ) ) )
4847adantr 451 . . . . 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 25 . . . 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 79 . . 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 420 . 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 578 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   ` cfv 5334  (class class class)co 5942   CHcch 21617   _|_cort 21618    vH chj 21621   0Hc0h 21623  HAtomscat 21653
This theorem is referenced by:  chirredlem2  23079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cc 8148  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904  ax-hilex 21687  ax-hfvadd 21688  ax-hvcom 21689  ax-hvass 21690  ax-hv0cl 21691  ax-hvaddid 21692  ax-hfvmul 21693  ax-hvmulid 21694  ax-hvmulass 21695  ax-hvdistr1 21696  ax-hvdistr2 21697  ax-hvmul0 21698  ax-hfi 21766  ax-his1 21769  ax-his2 21770  ax-his3 21771  ax-his4 21772  ax-hcompl 21889
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-omul 6568  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-acn 7662  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-cn 17057  df-cnp 17058  df-lm 17059  df-haus 17143  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cfil 18779  df-cau 18780  df-cmet 18781  df-grpo 20964  df-gid 20965  df-ginv 20966  df-gdiv 20967  df-ablo 21055  df-subgo 21075  df-vc 21210  df-nv 21256  df-va 21259  df-ba 21260  df-sm 21261  df-0v 21262  df-vs 21263  df-nmcv 21264  df-ims 21265  df-dip 21382  df-ssp 21406  df-ph 21499  df-cbn 21550  df-hnorm 21656  df-hba 21657  df-hvsub 21659  df-hlim 21660  df-hcau 21661  df-sh 21894  df-ch 21909  df-oc 21939  df-ch0 21940  df-shs 21995  df-span 21996  df-chj 21997  df-chsup 21998  df-pjh 22082  df-cv 22967  df-at 23026
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