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Theorem dicelvalN 38308
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)
Assertion
Ref Expression
dicelvalN (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐸(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)   𝑌(𝑔)

Proof of Theorem dicelvalN
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . 4 = (le‘𝐾)
2 dicval.a . . . 4 𝐴 = (Atoms‘𝐾)
3 dicval.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dicval.p . . . 4 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . . 4 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicval 38306 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
98eleq2d 2898 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)}))
10 vex 3498 . . . . . 6 𝑓 ∈ V
11 vex 3498 . . . . . 6 𝑠 ∈ V
1210, 11op1std 7693 . . . . 5 (𝑌 = ⟨𝑓, 𝑠⟩ → (1st𝑌) = 𝑓)
1310, 11op2ndd 7694 . . . . . 6 (𝑌 = ⟨𝑓, 𝑠⟩ → (2nd𝑌) = 𝑠)
1413fveq1d 6667 . . . . 5 (𝑌 = ⟨𝑓, 𝑠⟩ → ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1512, 14eqeq12d 2837 . . . 4 (𝑌 = ⟨𝑓, 𝑠⟩ → ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1613eleq1d 2897 . . . 4 (𝑌 = ⟨𝑓, 𝑠⟩ → ((2nd𝑌) ∈ 𝐸𝑠𝐸))
1715, 16anbi12d 632 . . 3 (𝑌 = ⟨𝑓, 𝑠⟩ → (((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
1817elopaba 5676 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸)))
199, 18syl6bb 289 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3495  cop 4567   class class class wbr 5059  {copab 5121   × cxp 5548  cfv 6350  crio 7107  1st c1st 7681  2nd c2nd 7682  lecple 16566  occoc 16567  Atomscatm 36393  LHypclh 37114  LTrncltrn 37231  TEndoctendo 37882  DIsoCcdic 38302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-1st 7683  df-2nd 7684  df-dic 38303
This theorem is referenced by:  dicelval2N  38312
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