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Theorem omlsilem 21977
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 21790 . . . . . . . . 9  |-  A  e. 
~H
4 omlsilem.1 . . . . . . . . . 10  |-  G  e.  SH
5 omlsilem.6 . . . . . . . . . 10  |-  B  e.  G
64, 5shelii 21790 . . . . . . . . 9  |-  B  e. 
~H
7 shocss 21861 . . . . . . . . . . 11  |-  ( G  e.  SH  ->  ( _|_ `  G )  C_  ~H )
84, 7ax-mp 8 . . . . . . . . . 10  |-  ( _|_ `  G )  C_  ~H
9 omlsilem.7 . . . . . . . . . 10  |-  C  e.  ( _|_ `  G
)
108, 9sselii 3178 . . . . . . . . 9  |-  C  e. 
~H
113, 6, 10hvsubaddi 21641 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
12 eqcom 2286 . . . . . . . 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 3178 . . . . . . . . 9  |-  B  e.  H
16 shsubcl 21796 . . . . . . . . 9  |-  ( ( H  e.  SH  /\  A  e.  H  /\  B  e.  H )  ->  ( A  -h  B
)  e.  H )
171, 2, 15, 16mp3an 1277 . . . . . . . 8  |-  ( A  -h  B )  e.  H
18 eleq1 2344 . . . . . . . 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 2348 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  C  e.  0H )
23 elin 3359 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  ( C  e.  H  /\  C  e.  ( _|_ `  G
) ) )
24 elch0 21829 . . . . . . . 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 5836 . . . 4  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  ( B  +h  0h ) )
29 ax-hvaddid 21580 . . . . 5  |-  ( B  e.  ~H  ->  ( B  +h  0h )  =  B )
306, 29ax-mp 8 . . . 4  |-  ( B  +h  0h )  =  B
3128, 30syl6eq 2332 . . 3  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  B )
3231, 5syl6eqel 2372 . 2  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  e.  G )
33 eleq1 2344 . 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 1623    e. wcel 1685    i^i cin 3152    C_ wss 3153   ` cfv 5221  (class class class)co 5820   ~Hchil 21495    +h cva 21496   0hc0v 21500    -h cmv 21501   SHcsh 21504   _|_cort 21506   0Hc0h 21511
This theorem is referenced by:  omlsii  21978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-hilex 21575  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvdistr2 21585  ax-hvmul0 21586  ax-hfi 21654  ax-his2 21658  ax-his3 21659
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-ltxr 8868  df-sub 9035  df-neg 9036  df-hvsub 21547  df-sh 21782  df-oc 21827  df-ch0 21828
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