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Theorem dia0eldmN 31852
Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dia0eldm.z  |-  .0.  =  ( 0. `  K )
dia0eldm.h  |-  H  =  ( LHyp `  K
)
dia0eldm.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
dia0eldmN  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  dom  I
)

Proof of Theorem dia0eldmN
StepHypRef Expression
1 hlop 30174 . . . 4  |-  ( K  e.  HL  ->  K  e.  OP )
21adantr 451 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  K  e.  OP )
3 eqid 2296 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
4 dia0eldm.z . . . 4  |-  .0.  =  ( 0. `  K )
53, 4op0cl 29996 . . 3  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
62, 5syl 15 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  ( Base `  K ) )
7 dia0eldm.h . . . 4  |-  H  =  ( LHyp `  K
)
83, 7lhpbase 30809 . . 3  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
9 eqid 2296 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
103, 9, 4op0le 29998 . . 3  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  ->  .0.  ( le `  K
) W )
111, 8, 10syl2an 463 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  ( le `  K ) W )
12 dia0eldm.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
133, 9, 7, 12diaeldm 31848 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  .0.  e.  dom  I 
<->  (  .0.  e.  (
Base `  K )  /\  .0.  ( le `  K ) W ) ) )
146, 11, 13mpbir2and 888 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  e.  dom  I
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   dom cdm 4705   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   OPcops 29984   HLchlt 30162   LHypclh 30795   DIsoAcdia 31840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-undef 6314  df-riota 6320  df-glb 14125  df-p0 14161  df-oposet 29988  df-ol 29990  df-oml 29991  df-hlat 30163  df-lhyp 30799  df-disoa 31841
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