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Theorem omlsi 22899
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 2442 . 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 2445 . 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 22750 . . . 4  |-  0H  e.  CH
53, 4keepel 3789 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  e.  CH
6 omls.2 . . . 4  |-  B  e.  SH
7 h0elsh 22751 . . . 4  |-  0H  e.  SH
86, 7keepel 3789 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  e.  SH
9 sseq1 3362 . . . . . 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 5721 . . . . . . . 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 3535 . . . . . . 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 2444 . . . . . 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 692 . . . . 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 3363 . . . . . 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 3528 . . . . . . 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 2444 . . . . . 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 692 . . . . 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 3362 . . . . . 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 5721 . . . . . . . 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 3535 . . . . . . 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 2444 . . . . . 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 692 . . . . 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 3363 . . . . . 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 3528 . . . . . . 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 2444 . . . . . 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 692 . . . . 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 3360 . . . . . 6  |-  0H  C_  0H
28 ocin 22791 . . . . . . 7  |-  ( 0H  e.  SH  ->  ( 0H  i^i  ( _|_ `  0H ) )  =  0H )
297, 28ax-mp 8 . . . . . 6  |-  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H
3027, 29pm3.2i 442 . . . . 5  |-  ( 0H  C_  0H  /\  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H )
3113, 17, 22, 26, 30elimhyp2v 3781 . . . 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 445 . . 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 449 . . 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 22898 . 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 3777 1  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3312    C_ wss 3313   ifcif 3732   ` cfv 5447   SHcsh 22424   CHcch 22425   _|_cort 22426   0Hc0h 22431
This theorem is referenced by:  pjomli  22930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-inf2 7589  ax-cc 8308  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060  ax-pre-sup 9061  ax-addf 9062  ax-mulf 9063  ax-hilex 22495  ax-hfvadd 22496  ax-hvcom 22497  ax-hvass 22498  ax-hv0cl 22499  ax-hvaddid 22500  ax-hfvmul 22501  ax-hvmulid 22502  ax-hvmulass 22503  ax-hvdistr1 22504  ax-hvdistr2 22505  ax-hvmul0 22506  ax-hfi 22574  ax-his1 22577  ax-his2 22578  ax-his3 22579  ax-his4 22580  ax-hcompl 22697
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-se 4535  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-isom 5456  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-omul 6722  df-er 6898  df-map 7013  df-pm 7014  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-fi 7409  df-sup 7439  df-oi 7472  df-card 7819  df-acn 7822  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-n0 10215  df-z 10276  df-uz 10482  df-q 10568  df-rp 10606  df-xneg 10703  df-xadd 10704  df-xmul 10705  df-ico 10915  df-icc 10916  df-fz 11037  df-fl 11195  df-seq 11317  df-exp 11376  df-cj 11897  df-re 11898  df-im 11899  df-sqr 12033  df-abs 12034  df-clim 12275  df-rlim 12276  df-rest 13643  df-topgen 13660  df-psmet 16687  df-xmet 16688  df-met 16689  df-bl 16690  df-mopn 16691  df-fbas 16692  df-fg 16693  df-top 16956  df-bases 16958  df-topon 16959  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-lm 17286  df-haus 17372  df-fil 17871  df-fm 17963  df-flim 17964  df-flf 17965  df-cfil 19201  df-cau 19202  df-cmet 19203  df-grpo 21772  df-gid 21773  df-ginv 21774  df-gdiv 21775  df-ablo 21863  df-subgo 21883  df-vc 22018  df-nv 22064  df-va 22067  df-ba 22068  df-sm 22069  df-0v 22070  df-vs 22071  df-nmcv 22072  df-ims 22073  df-ssp 22214  df-ph 22307  df-cbn 22358  df-hnorm 22464  df-hba 22465  df-hvsub 22467  df-hlim 22468  df-hcau 22469  df-sh 22702  df-ch 22717  df-oc 22747  df-ch0 22748
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