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Theorem dibelval2nd 41135
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 2735 . . . . 5 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibelval2nd.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 41127 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2825 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
109biimp3a 1468 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11 xp2nd 8046 . 2 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd𝑌) ∈ { 0 })
12 elsni 4648 . 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 1537  wcel 2106  {csn 4631   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  cres 5691  cfv 6563  2nd c2nd 8012  Basecbs 17245  lecple 17305  LHypclh 39967  LTrncltrn 40084  DIsoAcdia 41011  DIsoBcdib 41121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-2nd 8014  df-disoa 41012  df-dib 41122
This theorem is referenced by:  diblss  41153
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