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Theorem omlsi 21985
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 2291 . 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 2294 . 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 21836 . . . 4  |-  0H  e.  CH
53, 4keepel 3624 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  A ,  0H )  e.  CH
6 omls.2 . . . 4  |-  B  e.  SH
7 h0elsh 21837 . . . 4  |-  0H  e.  SH
86, 7keepel 3624 . . 3  |-  if ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H ) ,  B ,  0H )  e.  SH
9 sseq1 3201 . . . . . 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 5527 . . . . . . . 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 3372 . . . . . . 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 2293 . . . . . 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 3202 . . . . . 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 3365 . . . . . . 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 2293 . . . . . 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 3201 . . . . . 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 5527 . . . . . . . 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 3372 . . . . . . 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 2293 . . . . . 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 3202 . . . . . 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 3365 . . . . . . 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 2293 . . . . . 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 3199 . . . . . 6  |-  0H  C_  0H
28 ocin 21877 . . . . . . 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 3616 . . . 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 21984 . 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 3612 1  |-  ( ( A  C_  B  /\  ( B  i^i  ( _|_ `  A ) )  =  0H )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   ifcif 3567   ` cfv 5257   SHcsh 21510   CHcch 21511   _|_cort 21512   0Hc0h 21517
This theorem is referenced by:  pjomli  22016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cc 8063  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666  ax-hcompl 21783
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-n0 9968  df-z 10027  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ico 10664  df-icc 10665  df-fz 10785  df-fl 10927  df-seq 11049  df-exp 11107  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-rest 13329  df-topgen 13346  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-top 16638  df-bases 16640  df-topon 16641  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lm 16961  df-haus 17045  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-cfil 18683  df-cau 18684  df-cmet 18685  df-grpo 20860  df-gid 20861  df-ginv 20862  df-gdiv 20863  df-ablo 20951  df-subgo 20971  df-vc 21104  df-nv 21150  df-va 21153  df-ba 21154  df-sm 21155  df-0v 21156  df-vs 21157  df-nmcv 21158  df-ims 21159  df-ssp 21300  df-ph 21393  df-cbn 21444  df-hnorm 21550  df-hba 21551  df-hvsub 21553  df-hlim 21554  df-hcau 21555  df-sh 21788  df-ch 21803  df-oc 21833  df-ch0 21834
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