HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  omlsi Structured version   Unicode version

Theorem omlsi 25986
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 2466 . 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 2477 . 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 25837 . . . 4  |-  0H  e.  CH
53, 4keepel 4002 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  e.  CH
6 omls.2 . . . 4  |-  B  e.  SH
7 h0elsh 25838 . . . 4  |-  0H  e.  SH
86, 7keepel 4002 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  e.  SH
9 sseq1 3520 . . . . . 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 5859 . . . . . . . 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 3695 . . . . . . 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 2464 . . . . . 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 710 . . . . 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 3521 . . . . . 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 3688 . . . . . . 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 2464 . . . . . 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 710 . . . . 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 3520 . . . . . 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 5859 . . . . . . . 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 3695 . . . . . . 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 2464 . . . . . 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 710 . . . . 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 3521 . . . . . 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 3688 . . . . . . 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 2464 . . . . . 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 710 . . . . 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 3518 . . . . . 6  |-  0H  C_  0H
28 ocin 25878 . . . . . . 7  |-  ( 0H  e.  SH  ->  ( 0H  i^i  ( _|_ `  0H ) )  =  0H )
297, 28ax-mp 5 . . . . . 6  |-  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H
3027, 29pm3.2i 455 . . . . 5  |-  ( 0H  C_  0H  /\  ( 0H 
i^i  ( _|_ `  0H ) )  =  0H )
3113, 17, 22, 26, 30elimhyp2v 3994 . . . 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 458 . . 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 462 . . 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 25985 . 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 3990 1  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    i^i cin 3470    C_ wss 3471   ifcif 3934   ` cfv 5581   SHcsh 25509   CHcch 25510   _|_cort 25511   0Hc0h 25516
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cc 8806  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563  ax-hilex 25580  ax-hfvadd 25581  ax-hvcom 25582  ax-hvass 25583  ax-hv0cl 25584  ax-hvaddid 25585  ax-hfvmul 25586  ax-hvmulid 25587  ax-hvmulass 25588  ax-hvdistr1 25589  ax-hvdistr2 25590  ax-hvmul0 25591  ax-hfi 25660  ax-his1 25663  ax-his2 25664  ax-his3 25665  ax-his4 25666  ax-hcompl 25783
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-acn 8314  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ico 11526  df-icc 11527  df-fz 11664  df-fl 11888  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-clim 13262  df-rlim 13263  df-rest 14669  df-topgen 14690  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-top 19161  df-bases 19163  df-topon 19164  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lm 19491  df-haus 19577  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-cfil 21424  df-cau 21425  df-cmet 21426  df-grpo 24857  df-gid 24858  df-ginv 24859  df-gdiv 24860  df-ablo 24948  df-subgo 24968  df-vc 25103  df-nv 25149  df-va 25152  df-ba 25153  df-sm 25154  df-0v 25155  df-vs 25156  df-nmcv 25157  df-ims 25158  df-ssp 25299  df-ph 25392  df-cbn 25443  df-hnorm 25549  df-hba 25550  df-hvsub 25552  df-hlim 25553  df-hcau 25554  df-sh 25788  df-ch 25803  df-oc 25834  df-ch0 25835
This theorem is referenced by:  pjomli  26017
  Copyright terms: Public domain W3C validator