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Theorem omlsilem 22857
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 22670 . . . . . . . . 9  |-  A  e. 
~H
4 omlsilem.1 . . . . . . . . . 10  |-  G  e.  SH
5 omlsilem.6 . . . . . . . . . 10  |-  B  e.  G
64, 5shelii 22670 . . . . . . . . 9  |-  B  e. 
~H
7 shocss 22741 . . . . . . . . . . 11  |-  ( G  e.  SH  ->  ( _|_ `  G )  C_  ~H )
84, 7ax-mp 8 . . . . . . . . . 10  |-  ( _|_ `  G )  C_  ~H
9 omlsilem.7 . . . . . . . . . 10  |-  C  e.  ( _|_ `  G
)
108, 9sselii 3305 . . . . . . . . 9  |-  C  e. 
~H
113, 6, 10hvsubaddi 22521 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
12 eqcom 2406 . . . . . . . 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 3305 . . . . . . . . 9  |-  B  e.  H
16 shsubcl 22676 . . . . . . . . 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 2464 . . . . . . . 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 2468 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  C  e.  0H )
23 elin 3490 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  ( C  e.  H  /\  C  e.  ( _|_ `  G
) ) )
24 elch0 22709 . . . . . . . 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 6056 . . . 4  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  ( B  +h  0h ) )
29 ax-hvaddid 22460 . . . . 5  |-  ( B  e.  ~H  ->  ( B  +h  0h )  =  B )
306, 29ax-mp 8 . . . 4  |-  ( B  +h  0h )  =  B
3128, 30syl6eq 2452 . . 3  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  B )
3231, 5syl6eqel 2492 . 2  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  e.  G )
33 eleq1 2464 . 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 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   ` cfv 5413  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381   SHcsh 22384   _|_cort 22386   0Hc0h 22391
This theorem is referenced by:  omlsii  22858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his2 22538  ax-his3 22539
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-hvsub 22427  df-sh 22662  df-oc 22707  df-ch0 22708
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