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Theorem dicelval2N 35489
 Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
dicval.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicval2.g 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
Assertion
Ref Expression
dicelval2N (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸))))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐸(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)   𝑌(𝑔)

Proof of Theorem dicelval2N
StepHypRef Expression
1 dicval.l . . 3 = (le‘𝐾)
2 dicval.a . . 3 𝐴 = (Atoms‘𝐾)
3 dicval.h . . 3 𝐻 = (LHyp‘𝐾)
4 dicval.p . . 3 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . 3 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . 3 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicelvalN 35485 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))))
9 dicval2.g . . . . . 6 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
109fveq2i 6106 . . . . 5 ((2nd𝑌)‘𝐺) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄))
1110eqeq2i 2622 . . . 4 ((1st𝑌) = ((2nd𝑌)‘𝐺) ↔ (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1211anbi1i 727 . . 3 (((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸) ↔ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))
1312anbi2i 726 . 2 ((𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸)) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸)))
148, 13syl6bbr 277 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   × cxp 5036  ‘cfv 5804  ℩crio 6510  1st c1st 7057  2nd c2nd 7058  lecple 15775  occoc 15776  Atomscatm 33568  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  DIsoCcdic 35479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-1st 7059  df-2nd 7060  df-dic 35480 This theorem is referenced by: (None)
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