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Theorem dibelval2nd 35960
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 2621 . . . . 5 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
7 dibelval2nd.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dibval2 35952 . . . 4 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
98eleq2d 2684 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 })))
109biimp3a 1429 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → 𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }))
11 xp2nd 7159 . 2 (𝑌 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑋) × { 0 }) → (2nd𝑌) ∈ { 0 })
12 elsni 4172 . 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 384  w3a 1036   = wceq 1480  wcel 1987  {csn 4155   class class class wbr 4623  cmpt 4683   I cid 4994   × cxp 5082  cres 5086  cfv 5857  2nd c2nd 7127  Basecbs 15800  lecple 15888  LHypclh 34789  LTrncltrn 34906  DIsoAcdia 35836  DIsoBcdib 35946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-2nd 7129  df-disoa 35837  df-dib 35947
This theorem is referenced by:  diblss  35978
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