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Theorem dicval2 36287
Description: The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)
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
dicval2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
Distinct variable groups:   𝑓,𝑔,𝑠,𝐾   𝑇,𝑔   𝑓,𝑊,𝑔,𝑠   𝑓,𝐸,𝑠   𝑃,𝑓   𝑄,𝑓,𝑔,𝑠   𝑇,𝑓
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠)   𝑃(𝑔,𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐺(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   𝐼(𝑓,𝑔,𝑠)   (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)

Proof of Theorem dicval2
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, 7dicval 36284 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
9 dicval2.g . . . . . 6 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
109fveq2i 6181 . . . . 5 (𝑠𝐺) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))
1110eqeq2i 2632 . . . 4 (𝑓 = (𝑠𝐺) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1211anbi1i 730 . . 3 ((𝑓 = (𝑠𝐺) ∧ 𝑠𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸))
1312opabbii 4708 . 2 {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)}
148, 13syl6eqr 2672 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988   class class class wbr 4644  {copab 4703  cfv 5876  crio 6595  lecple 15929  occoc 15930  Atomscatm 34369  LHypclh 35089  LTrncltrn 35206  TEndoctendo 35859  DIsoCcdic 36280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-dic 36281
This theorem is referenced by:  dicelval3  36288  diclspsn  36302  dih1dimatlem  36437
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