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Theorem atexch 25720
Description: The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862. Also Definition 3.4-3(b) in [MegPav2000] p. 2345 (PDF p. 8) (use atnemeq0 25716 to obtain atom inequality). (Contributed by NM, 27-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
atexch  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  C  C_  ( A  vH  B
) ) )

Proof of Theorem atexch
StepHypRef Expression
1 atelch 25683 . . . . . 6  |-  ( C  e. HAtoms  ->  C  e.  CH )
2 chub2 24846 . . . . . . 7  |-  ( ( C  e.  CH  /\  A  e.  CH )  ->  C  C_  ( A  vH  C ) )
32ancoms 450 . . . . . 6  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  C  C_  ( A  vH  C ) )
41, 3sylan2 471 . . . . 5  |-  ( ( A  e.  CH  /\  C  e. HAtoms )  ->  C 
C_  ( A  vH  C ) )
543adant2 1002 . . . 4  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  C  C_  ( A  vH  C ) )
65adantr 462 . . 3  |-  ( ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  /\  ( B  C_  ( A  vH  C
)  /\  ( A  i^i  B )  =  0H ) )  ->  C  C_  ( A  vH  C
) )
7 cvp 25714 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e. HAtoms )  ->  ( ( A  i^i  B
)  =  0H  <->  A  <oH  ( A  vH  B ) ) )
8 atelch 25683 . . . . . . . . . . 11  |-  ( B  e. HAtoms  ->  B  e.  CH )
9 chjcl 24695 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  vH  B
)  e.  CH )
108, 9sylan2 471 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  vH  B )  e.  CH )
11 cvpss 25624 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  ( A  vH  B )  e.  CH )  -> 
( A  <oH  ( A  vH  B )  ->  A  C.  ( A  vH  B ) ) )
1210, 11syldan 467 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e. HAtoms )  ->  ( A  <oH  ( A  vH  B )  ->  A  C.  ( A  vH  B
) ) )
137, 12sylbid 215 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e. HAtoms )  ->  ( ( A  i^i  B
)  =  0H  ->  A 
C.  ( A  vH  B ) ) )
14133adant3 1003 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( A  i^i  B )  =  0H  ->  A  C.  ( A  vH  B ) ) )
1514adantld 464 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  A  C.  ( A  vH  B
) ) )
16 id 22 . . . . . . . 8  |-  ( A  e.  CH  ->  A  e.  CH )
17 chub1 24845 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  A  C_  ( A  vH  C ) )
18173adant2 1002 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  A  C_  ( A  vH  C
) )
1918a1d 25 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  C_  ( A  vH  C )  ->  A  C_  ( A  vH  C
) ) )
2019ancrd 551 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  C_  ( A  vH  C )  ->  ( A  C_  ( A  vH  C )  /\  B  C_  ( A  vH  C
) ) ) )
21 chjcl 24695 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  vH  C
)  e.  CH )
22213adant2 1002 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  vH  C )  e. 
CH )
23 chlub 24847 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  ( A  vH  C )  e. 
CH )  ->  (
( A  C_  ( A  vH  C )  /\  B  C_  ( A  vH  C ) )  <->  ( A  vH  B )  C_  ( A  vH  C ) ) )
2422, 23syld3an3 1258 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  C_  ( A  vH  C )  /\  B  C_  ( A  vH  C ) )  <->  ( A  vH  B )  C_  ( A  vH  C ) ) )
2520, 24sylibd 214 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( B  C_  ( A  vH  C )  ->  ( A  vH  B )  C_  ( A  vH  C ) ) )
2616, 8, 1, 25syl3an 1255 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( B  C_  ( A  vH  C
)  ->  ( A  vH  B )  C_  ( A  vH  C ) ) )
2726adantrd 465 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  ( A  vH  B )  C_  ( A  vH  C ) ) )
2815, 27jcad 530 . . . . 5  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  ( A  C.  ( A  vH  B )  /\  ( A  vH  B )  C_  ( A  vH  C ) ) ) )
2928imp 429 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  /\  ( B  C_  ( A  vH  C
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  C.  ( A  vH  B )  /\  ( A  vH  B )  C_  ( A  vH  C ) ) )
30 simp1 983 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  A  e.  CH )
3193adant3 1003 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  vH  B )  e. 
CH )
3230, 22, 313jca 1163 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  e.  CH  /\  ( A  vH  C )  e. 
CH  /\  ( A  vH  B )  e.  CH ) )
3316, 8, 1, 32syl3an 1255 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( A  e.  CH  /\  ( A  vH  C )  e. 
CH  /\  ( A  vH  B )  e.  CH ) )
3414, 26anim12d 560 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( (
( A  i^i  B
)  =  0H  /\  B  C_  ( A  vH  C ) )  -> 
( A  C.  ( A  vH  B )  /\  ( A  vH  B ) 
C_  ( A  vH  C ) ) ) )
3534ancomsd 451 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  ( A  C.  ( A  vH  B )  /\  ( A  vH  B )  C_  ( A  vH  C ) ) ) )
36 psssstr 3459 . . . . . . . 8  |-  ( ( A  C.  ( A  vH  B )  /\  ( A  vH  B )  C_  ( A  vH  C ) )  ->  A  C.  ( A  vH  C ) )
3735, 36syl6 33 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  A  C.  ( A  vH  C
) ) )
38 chcv2 25695 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e. HAtoms )  ->  ( A  C.  ( A  vH  C )  <->  A  <oH  ( A  vH  C ) ) )
39383adant2 1002 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( A  C.  ( A  vH  C
)  <->  A  <oH  ( A  vH  C ) ) )
4037, 39sylibd 214 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  A  <oH  ( A  vH  C
) ) )
41 cvnbtwn2 25626 . . . . . 6  |-  ( ( A  e.  CH  /\  ( A  vH  C )  e.  CH  /\  ( A  vH  B )  e. 
CH )  ->  ( A  <oH  ( A  vH  C )  ->  (
( A  C.  ( A  vH  B )  /\  ( A  vH  B ) 
C_  ( A  vH  C ) )  -> 
( A  vH  B
)  =  ( A  vH  C ) ) ) )
4233, 40, 41sylsyld 56 . . . . 5  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  (
( A  C.  ( A  vH  B )  /\  ( A  vH  B ) 
C_  ( A  vH  C ) )  -> 
( A  vH  B
)  =  ( A  vH  C ) ) ) )
4342imp 429 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  /\  ( B  C_  ( A  vH  C
)  /\  ( A  i^i  B )  =  0H ) )  ->  (
( A  C.  ( A  vH  B )  /\  ( A  vH  B ) 
C_  ( A  vH  C ) )  -> 
( A  vH  B
)  =  ( A  vH  C ) ) )
4429, 43mpd 15 . . 3  |-  ( ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  /\  ( B  C_  ( A  vH  C
)  /\  ( A  i^i  B )  =  0H ) )  ->  ( A  vH  B )  =  ( A  vH  C
) )
456, 44sseqtr4d 3390 . 2  |-  ( ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  /\  ( B  C_  ( A  vH  C
)  /\  ( A  i^i  B )  =  0H ) )  ->  C  C_  ( A  vH  B
) )
4645ex 434 1  |-  ( ( A  e.  CH  /\  B  e. HAtoms  /\  C  e. HAtoms
)  ->  ( ( B  C_  ( A  vH  C )  /\  ( A  i^i  B )  =  0H )  ->  C  C_  ( A  vH  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325    C. wpss 3326   class class class wbr 4289  (class class class)co 6090   CHcch 24266    vH chj 24270   0Hc0h 24272    <oH ccv 24301  HAtomscat 24302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvmulass 24344  ax-hvdistr1 24345  ax-hvdistr2 24346  ax-hvmul0 24347  ax-hfi 24416  ax-his1 24419  ax-his2 24420  ax-his3 24421  ax-his4 24422  ax-hcompl 24539
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-cn 18790  df-cnp 18791  df-lm 18792  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cfil 20725  df-cau 20726  df-cmet 20727  df-grpo 23613  df-gid 23614  df-ginv 23615  df-gdiv 23616  df-ablo 23704  df-subgo 23724  df-vc 23859  df-nv 23905  df-va 23908  df-ba 23909  df-sm 23910  df-0v 23911  df-vs 23912  df-nmcv 23913  df-ims 23914  df-dip 24031  df-ssp 24055  df-ph 24148  df-cbn 24199  df-hnorm 24305  df-hba 24306  df-hvsub 24308  df-hlim 24309  df-hcau 24310  df-sh 24544  df-ch 24559  df-oc 24590  df-ch0 24591  df-shs 24646  df-span 24647  df-chj 24648  df-chsup 24649  df-pjh 24733  df-cv 25618  df-at 25677
This theorem is referenced by:  atomli  25721  atcvatlem  25724  atcvat4i  25736  mdsymlem3  25744  mdsymlem5  25746
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