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Theorem omlsilem 22095
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 21908 . . . . . . . . 9  |-  A  e. 
~H
4 omlsilem.1 . . . . . . . . . 10  |-  G  e.  SH
5 omlsilem.6 . . . . . . . . . 10  |-  B  e.  G
64, 5shelii 21908 . . . . . . . . 9  |-  B  e. 
~H
7 shocss 21979 . . . . . . . . . . 11  |-  ( G  e.  SH  ->  ( _|_ `  G )  C_  ~H )
84, 7ax-mp 8 . . . . . . . . . 10  |-  ( _|_ `  G )  C_  ~H
9 omlsilem.7 . . . . . . . . . 10  |-  C  e.  ( _|_ `  G
)
108, 9sselii 3253 . . . . . . . . 9  |-  C  e. 
~H
113, 6, 10hvsubaddi 21759 . . . . . . . 8  |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
12 eqcom 2360 . . . . . . . 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 3253 . . . . . . . . 9  |-  B  e.  H
16 shsubcl 21914 . . . . . . . . 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 2418 . . . . . . . 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 2422 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  C  e.  0H )
23 elin 3434 . . . . . . . 8  |-  ( C  e.  ( H  i^i  ( _|_ `  G ) )  <->  ( C  e.  H  /\  C  e.  ( _|_ `  G
) ) )
24 elch0 21947 . . . . . . . 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 5961 . . . 4  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  ( B  +h  0h ) )
29 ax-hvaddid 21698 . . . . 5  |-  ( B  e.  ~H  ->  ( B  +h  0h )  =  B )
306, 29ax-mp 8 . . . 4  |-  ( B  +h  0h )  =  B
3128, 30syl6eq 2406 . . 3  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  =  B )
3231, 5syl6eqel 2446 . 2  |-  ( A  =  ( B  +h  C )  ->  ( B  +h  C )  e.  G )
33 eleq1 2418 . 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 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   ` cfv 5337  (class class class)co 5945   ~Hchil 21613    +h cva 21614   0hc0v 21618    -h cmv 21619   SHcsh 21622   _|_cort 21624   0Hc0h 21629
This theorem is referenced by:  omlsii  22096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-hilex 21693  ax-hfvadd 21694  ax-hvcom 21695  ax-hvass 21696  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvdistr2 21703  ax-hvmul0 21704  ax-hfi 21772  ax-his2 21776  ax-his3 21777
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-ltxr 8962  df-sub 9129  df-neg 9130  df-hvsub 21665  df-sh 21900  df-oc 21945  df-ch0 21946
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