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Theorem dicelval2N 38933
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 38929 . 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 6720 . . . . 5 ((2nd𝑌)‘𝐺) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄))
1110eqeq2i 2750 . . . 4 ((1st𝑌) = ((2nd𝑌)‘𝐺) ↔ (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1211anbi1i 627 . . 3 (((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸) ↔ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))
1312anbi2i 626 . 2 ((𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸)) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸)))
148, 13bitr4di 292 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 209  wa 399   = wceq 1543  wcel 2110  Vcvv 3408   class class class wbr 5053   × cxp 5549  cfv 6380  crio 7169  1st c1st 7759  2nd c2nd 7760  lecple 16809  occoc 16810  Atomscatm 37014  LHypclh 37735  LTrncltrn 37852  TEndoctendo 38503  DIsoCcdic 38923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-1st 7761  df-2nd 7762  df-dic 38924
This theorem is referenced by: (None)
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