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Theorem omlsilem 22294
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 22107 . . . . . . . . 9  |-  A  e. 
~H
4 omlsilem.1 . . . . . . . . . 10  |-  G  e.  SH
5 omlsilem.6 . . . . . . . . . 10  |-  B  e.  G
64, 5shelii 22107 . . . . . . . . 9  |-  B  e. 
~H
7 shocss 22178 . . . . . . . . . . 11  |-  ( G  e.  SH  ->  ( _|_ `  G )  C_  ~H )
84, 7ax-mp 8 . . . . . . . . . 10  |-  ( _|_ `  G )  C_  ~H
9 omlsilem.7 . . . . . . . . . 10  |-  C  e.  ( _|_ `  G
)
108, 9sselii 3263 . . . . . . . . 9  |-  C  e. 
~H
113, 6, 10hvsubaddi 21958 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
12 eqcom 2368 . . . . . . . 8  |-  ( ( B  +h  C )  =  A  <->  A  =  ( B  +h  C
) )
1311, 12bitri 240 . . . . . . 7  |-  ( ( A  -h  B )  =  C  <->  A  =  ( B  +h  C
) )
14 omlsilem.3 . . . . . . . . . 10  |-  G  C_  H
1514, 5sselii 3263 . . . . . . . . 9  |-  B  e.  H
16 shsubcl 22113 . . . . . . . . 9  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  -h  B
)  e.  H )
171, 2, 15, 16mp3an 1278 . . . . . . . 8  |-  ( A  -h  B )  e.  H
18 eleq1 2426 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  ->  (
( A  -h  B
)  e.  H  <->  C  e.  H ) )
1917, 18mpbii 202 . . . . . . 7  |-  ( ( A  -h  B )  =  C  ->  C  e.  H )
2013, 19sylbir 204 . . . . . 6  |-  ( A  =  ( B  +h  C )  ->  C  e.  H )
21 omlsilem.4 . . . . . . . . 9  |-  ( H  i^i  ( _|_ `  G
) )  =  0H
2221eleq2i 2430 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  C  e.  0H )
23 elin 3446 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  ( C  e.  H  /\  C  e.  ( _|_ `  G
) ) )
24 elch0 22146 . . . . . . . 8  |-  ( C  e.  0H  <->  C  =  0h )
2522, 23, 243bitr3i 266 . . . . . . 7  |-  ( ( C  e.  H  /\  C  e.  ( _|_ `  G ) )  <->  C  =  0h )
2625biimpi 186 . . . . . 6  |-  ( ( C  e.  H  /\  C  e.  ( _|_ `  G ) )  ->  C  =  0h )
2720, 9, 26sylancl 643 . . . . 5  |-  ( A  =  ( B  +h  C )  ->  C  =  0h )
2827oveq2d 5997 . . . 4  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  ( B  +h  0h ) )
29 ax-hvaddid 21897 . . . . 5  |-  ( B  e.  ~H  ->  ( B  +h  0h )  =  B )
306, 29ax-mp 8 . . . 4  |-  ( B  +h  0h )  =  B
3128, 30syl6eq 2414 . . 3  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  B )
3231, 5syl6eqel 2454 . 2  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  e.  G )
33 eleq1 2426 . 2  |-  ( A  =  ( B  +h  C )  ->  ( A  e.  G  <->  ( B  +h  C )  e.  G
) )
3432, 33mpbird 223 1  |-  ( A  =  ( B  +h  C )  ->  A  e.  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    i^i cin 3237    C_ wss 3238   ` cfv 5358  (class class class)co 5981   ~Hchil 21812    +h cva 21813   0hc0v 21817    -h cmv 21818   SHcsh 21821   _|_cort 21823   0Hc0h 21828
This theorem is referenced by:  omlsii  22295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-hilex 21892  ax-hfvadd 21893  ax-hvcom 21894  ax-hvass 21895  ax-hv0cl 21896  ax-hvaddid 21897  ax-hfvmul 21898  ax-hvmulid 21899  ax-hvdistr2 21902  ax-hvmul0 21903  ax-hfi 21971  ax-his2 21975  ax-his3 21976
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019  df-sub 9186  df-neg 9187  df-hvsub 21864  df-sh 22099  df-oc 22144  df-ch0 22145
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