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Theorem omlsilem 22904
Description: Lemma for orthomodular law in the Hilbert lattice. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlsilem.1  |-  G  e.  SH
omlsilem.2  |-  H  e.  SH
omlsilem.3  |-  G  C_  H
omlsilem.4  |-  ( H  i^i  ( _|_ `  G
) )  =  0H
omlsilem.5  |-  A  e.  H
omlsilem.6  |-  B  e.  G
omlsilem.7  |-  C  e.  ( _|_ `  G
)
Assertion
Ref Expression
omlsilem  |-  ( A  =  ( B  +h  C )  ->  A  e.  G )

Proof of Theorem omlsilem
StepHypRef Expression
1 omlsilem.2 . . . . . . . . . 10  |-  H  e.  SH
2 omlsilem.5 . . . . . . . . . 10  |-  A  e.  H
31, 2shelii 22717 . . . . . . . . 9  |-  A  e. 
~H
4 omlsilem.1 . . . . . . . . . 10  |-  G  e.  SH
5 omlsilem.6 . . . . . . . . . 10  |-  B  e.  G
64, 5shelii 22717 . . . . . . . . 9  |-  B  e. 
~H
7 shocss 22788 . . . . . . . . . . 11  |-  ( G  e.  SH  ->  ( _|_ `  G )  C_  ~H )
84, 7ax-mp 8 . . . . . . . . . 10  |-  ( _|_ `  G )  C_  ~H
9 omlsilem.7 . . . . . . . . . 10  |-  C  e.  ( _|_ `  G
)
108, 9sselii 3345 . . . . . . . . 9  |-  C  e. 
~H
113, 6, 10hvsubaddi 22568 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
12 eqcom 2438 . . . . . . . 8  |-  ( ( B  +h  C )  =  A  <->  A  =  ( B  +h  C
) )
1311, 12bitri 241 . . . . . . 7  |-  ( ( A  -h  B )  =  C  <->  A  =  ( B  +h  C
) )
14 omlsilem.3 . . . . . . . . . 10  |-  G  C_  H
1514, 5sselii 3345 . . . . . . . . 9  |-  B  e.  H
16 shsubcl 22723 . . . . . . . . 9  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  -h  B
)  e.  H )
171, 2, 15, 16mp3an 1279 . . . . . . . 8  |-  ( A  -h  B )  e.  H
18 eleq1 2496 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  ->  (
( A  -h  B
)  e.  H  <->  C  e.  H ) )
1917, 18mpbii 203 . . . . . . 7  |-  ( ( A  -h  B )  =  C  ->  C  e.  H )
2013, 19sylbir 205 . . . . . 6  |-  ( A  =  ( B  +h  C )  ->  C  e.  H )
21 omlsilem.4 . . . . . . . . 9  |-  ( H  i^i  ( _|_ `  G
) )  =  0H
2221eleq2i 2500 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  C  e.  0H )
23 elin 3530 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  ( C  e.  H  /\  C  e.  ( _|_ `  G
) ) )
24 elch0 22756 . . . . . . . 8  |-  ( C  e.  0H  <->  C  =  0h )
2522, 23, 243bitr3i 267 . . . . . . 7  |-  ( ( C  e.  H  /\  C  e.  ( _|_ `  G ) )  <->  C  =  0h )
2625biimpi 187 . . . . . 6  |-  ( ( C  e.  H  /\  C  e.  ( _|_ `  G ) )  ->  C  =  0h )
2720, 9, 26sylancl 644 . . . . 5  |-  ( A  =  ( B  +h  C )  ->  C  =  0h )
2827oveq2d 6097 . . . 4  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  ( B  +h  0h ) )
29 ax-hvaddid 22507 . . . . 5  |-  ( B  e.  ~H  ->  ( B  +h  0h )  =  B )
306, 29ax-mp 8 . . . 4  |-  ( B  +h  0h )  =  B
3128, 30syl6eq 2484 . . 3  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  B )
3231, 5syl6eqel 2524 . 2  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  e.  G )
33 eleq1 2496 . 2  |-  ( A  =  ( B  +h  C )  ->  ( A  e.  G  <->  ( B  +h  C )  e.  G
) )
3432, 33mpbird 224 1  |-  ( A  =  ( B  +h  C )  ->  A  e.  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   ~Hchil 22422    +h cva 22423   0hc0v 22427    -h cmv 22428   SHcsh 22431   _|_cort 22433   0Hc0h 22438
This theorem is referenced by:  omlsii  22905
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-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his2 22585  ax-his3 22586
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 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293  df-neg 9294  df-hvsub 22474  df-sh 22709  df-oc 22754  df-ch0 22755
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