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Theorem dibelval1st1 36939
 Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1st1.b 𝐵 = (Base‘𝐾)
dibelval1st1.l = (le‘𝐾)
dibelval1st1.h 𝐻 = (LHyp‘𝐾)
dibelval1st1.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st1.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval1st1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ 𝑇)

Proof of Theorem dibelval1st1
StepHypRef Expression
1 dibelval1st1.b . . 3 𝐵 = (Base‘𝐾)
2 dibelval1st1.l . . 3 = (le‘𝐾)
3 dibelval1st1.h . . 3 𝐻 = (LHyp‘𝐾)
4 eqid 2758 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5 dibelval1st1.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dibelval1st 36938 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7 dibelval1st1.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
81, 2, 3, 7, 4diael 36832 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (1st𝑌) ∈ 𝑇)
96, 8syld3an3 1516 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ 𝑇)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137   class class class wbr 4802  ‘cfv 6047  1st c1st 7329  Basecbs 16057  lecple 16148  LHypclh 35771  LTrncltrn 35888  DIsoAcdia 36817  DIsoBcdib 36927 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-1st 7331  df-disoa 36818  df-dib 36928 This theorem is referenced by:  diblss  36959
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