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Theorem dibelval2nd 41155
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval2nd.b 𝐵 = (Base‘𝐾)
dibelval2nd.l = (le‘𝐾)
dibelval2nd.h 𝐻 = (LHyp‘𝐾)
dibelval2nd.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval2nd.o 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dibelval2nd.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval2nd (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (2nd𝑌) = 0 )
Distinct variable groups:   𝑓,𝐾   𝑓,𝑊
Allowed substitution hints:   𝐵(𝑓)   𝑇(𝑓)   𝐻(𝑓)   𝐼(𝑓)   (𝑓)   𝑉(𝑓)   𝑋(𝑓)   𝑌(𝑓)   0 (𝑓)

Proof of Theorem dibelval2nd
StepHypRef Expression
1 dibelval2nd.b . . . . 5 𝐵 = (Base‘𝐾)
2 dibelval2nd.l . . . . 5 = (le‘𝐾)
3 dibelval2nd.h . . . . 5 𝐻 = (LHyp‘𝐾)
4 dibelval2nd.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 dibelval2nd.o . . . . 5 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
6 eqid 2736 . . . . 5 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibelval2nd.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 41147 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2826 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
109biimp3a 1470 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11 xp2nd 8048 . 2 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd𝑌) ∈ { 0 })
12 elsni 4642 . 2 ((2nd𝑌) ∈ { 0 } → (2nd𝑌) = 0 )
1310, 11, 123syl 18 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (2nd𝑌) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  {csn 4625   class class class wbr 5142  cmpt 5224   I cid 5576   × cxp 5682  cres 5686  cfv 6560  2nd c2nd 8014  Basecbs 17248  lecple 17305  LHypclh 39987  LTrncltrn 40104  DIsoAcdia 41031  DIsoBcdib 41141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-2nd 8016  df-disoa 41032  df-dib 41142
This theorem is referenced by:  diblss  41173
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