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Theorem omlsi 21999
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 2302 . 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 2305 . 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 21850 . . . 4  |-  0H  e.  CH
53, 4keepel 3635 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  e.  CH
6 omls.2 . . . 4  |-  B  e.  SH
7 h0elsh 21851 . . . 4  |-  0H  e.  SH
86, 7keepel 3635 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  e.  SH
9 sseq1 3212 . . . . . 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 5541 . . . . . . . 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 3383 . . . . . . 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 2304 . . . . . 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 3213 . . . . . 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 3376 . . . . . . 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 2304 . . . . . 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 3212 . . . . . 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 5541 . . . . . . . 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 3383 . . . . . . 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 2304 . . . . . 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 3213 . . . . . 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 3376 . . . . . . 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 2304 . . . . . 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 3210 . . . . . 6  |-  0H  C_  0H
28 ocin 21891 . . . . . . 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 3627 . . . 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 21998 . 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 3623 1  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ifcif 3578   ` cfv 5271   SHcsh 21524   CHcch 21525   _|_cort 21526   0Hc0h 21531
This theorem is referenced by:  pjomli  22030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680  ax-hcompl 21797
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-icc 10679  df-fz 10799  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lm 16975  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-cfil 18697  df-cau 18698  df-cmet 18699  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-ssp 21314  df-ph 21407  df-cbn 21458  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-hcau 21569  df-sh 21802  df-ch 21817  df-oc 21847  df-ch0 21848
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