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Theorem omlsi 21983
Description: Subspace form of orthomodular law in the Hilbert lattice. Compare the orthomodular law in Theorem 2(ii) of [Kalmbach] p. 22. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
omls.1  |-  A  e. 
CH
omls.2  |-  B  e.  SH
Assertion
Ref Expression
omlsi  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )

Proof of Theorem omlsi
StepHypRef Expression
1 eqeq1 2289 . 2  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( A  =  B  <->  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H )  =  B
) )
2 eqeq2 2292 . 2  |-  ( B  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  =  B  <->  if (
( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H ) ) )
3 omls.1 . . . 4  |-  A  e. 
CH
4 h0elch 21834 . . . 4  |-  0H  e.  CH
53, 4keepel 3622 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  e.  CH
6 omls.2 . . . 4  |-  B  e.  SH
7 h0elsh 21835 . . . 4  |-  0H  e.  SH
86, 7keepel 3622 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  e.  SH
9 sseq1 3199 . . . . . 6  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( A  C_  B 
<->  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H )  C_  B ) )
10 fveq2 5525 . . . . . . . 8  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( _|_ `  A
)  =  ( _|_ `  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H ) ) )
1110ineq2d 3370 . . . . . . 7  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( B  i^i  ( _|_ `  A ) )  =  ( B  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) ) )
1211eqeq1d 2291 . . . . . 6  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( ( B  i^i  ( _|_ `  A
) )  =  0H  <->  ( B  i^i  ( _|_ `  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) )
139, 12anbi12d 691 . . . . 5  |-  ( A  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H )  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  B  /\  ( B  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) ) )
14 sseq2 3200 . . . . . 6  |-  ( B  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  B  <->  if (
( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H ) ) )
15 ineq1 3363 . . . . . . 7  |-  ( B  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( B  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  ( if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) ) )
1615eqeq1d 2291 . . . . . 6  |-  ( B  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( ( B  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) )
1714, 16anbi12d 691 . . . . 5  |-  ( B  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  C_  B  /\  ( B  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H )  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  /\  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) ) )
18 sseq1 3199 . . . . . 6  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( 0H  C_  0H 
<->  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H )  C_  0H ) )
19 fveq2 5525 . . . . . . . 8  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( _|_ `  0H )  =  ( _|_ `  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H ) ) )
2019ineq2d 3370 . . . . . . 7  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( 0H  i^i  ( _|_ `  0H ) )  =  ( 0H 
i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) ) )
2120eqeq1d 2291 . . . . . 6  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( ( 0H 
i^i  ( _|_ `  0H ) )  =  0H  <->  ( 0H  i^i  ( _|_ `  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) )
2218, 21anbi12d 691 . . . . 5  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  ->  ( ( 0H  C_  0H  /\  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H )  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  0H  /\  ( 0H  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) ) )
23 sseq2 3200 . . . . . 6  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  0H  <->  if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  A ,  0H )  C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H ) ) )
24 ineq1 3363 . . . . . . 7  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( 0H  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  ( if ( ( A 
C_  B  /\  ( B  i^i  ( _|_ `  A
) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) ) )
2524eqeq1d 2291 . . . . . 6  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( ( 0H 
i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) )
2623, 25anbi12d 691 . . . . 5  |-  ( 0H  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  ->  ( ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  C_  0H  /\  ( 0H  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H )  <->  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  /\  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H ) ) )
27 ssid 3197 . . . . . 6  |-  0H  C_  0H
28 ocin 21875 . . . . . . 7  |-  ( 0H  e.  SH  ->  ( 0H  i^i  ( _|_ `  0H ) )  =  0H )
297, 28ax-mp 8 . . . . . 6  |-  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H
3027, 29pm3.2i 441 . . . . 5  |-  ( 0H  C_  0H  /\  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H )
3113, 17, 22, 26, 30elimhyp2v 3614 . . . 4  |-  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  /\  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H )
3231simpli 444 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) 
C_  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )
3331simpri 448 . . 3  |-  ( if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  i^i  ( _|_ `  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H ) ) )  =  0H
345, 8, 32, 33omlsii 21982 . 2  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  =  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )
351, 2, 34dedth2v 3610 1  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ifcif 3565   ` cfv 5255   SHcsh 21508   CHcch 21509   _|_cort 21510   0Hc0h 21515
This theorem is referenced by:  pjomli  22014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-icc 10663  df-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lm 16959  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832
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