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Theorem dibelval1st2N 36940
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibelval1st2.b 𝐵 = (Base‘𝐾)
dibelval1st2.l = (le‘𝐾)
dibelval1st2.h 𝐻 = (LHyp‘𝐾)
dibelval1st2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st2.r 𝑅 = ((trL‘𝐾)‘𝑊)
dibelval1st2.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval1st2N (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (𝑅‘(1st𝑌)) 𝑋)

Proof of Theorem dibelval1st2N
StepHypRef Expression
1 dibelval1st2.b . . 3 𝐵 = (Base‘𝐾)
2 dibelval1st2.l . . 3 = (le‘𝐾)
3 dibelval1st2.h . . 3 𝐻 = (LHyp‘𝐾)
4 eqid 2758 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5 dibelval1st2.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dibelval1st 36938 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7 dibelval1st2.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 dibelval1st2.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
91, 2, 3, 7, 8, 4diatrl 36833 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st𝑌)) 𝑋)
106, 9syld3an3 1516 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (𝑅‘(1st𝑌)) 𝑋)
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  trLctrl 35946  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: (None)
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