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Theorem dicelvalN 36988
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 36986 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
98eleq2d 2836 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)}))
10 vex 3354 . . . . . 6 𝑓 ∈ V
11 vex 3354 . . . . . 6 𝑠 ∈ V
1210, 11op1std 7329 . . . . 5 (𝑌 = ⟨𝑓, 𝑠⟩ → (1st𝑌) = 𝑓)
1310, 11op2ndd 7330 . . . . . 6 (𝑌 = ⟨𝑓, 𝑠⟩ → (2nd𝑌) = 𝑠)
1413fveq1d 6335 . . . . 5 (𝑌 = ⟨𝑓, 𝑠⟩ → ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1512, 14eqeq12d 2786 . . . 4 (𝑌 = ⟨𝑓, 𝑠⟩ → ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ 𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1613eleq1d 2835 . . . 4 (𝑌 = ⟨𝑓, 𝑠⟩ → ((2nd𝑌) ∈ 𝐸𝑠𝐸))
1715, 16anbi12d 616 . . 3 (𝑌 = ⟨𝑓, 𝑠⟩ → (((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸) ↔ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
1817elopaba 5370 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸)))
199, 18syl6bb 276 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 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cop 4323   class class class wbr 4787  {copab 4847   × cxp 5248  cfv 6030  crio 6756  1st c1st 7317  2nd c2nd 7318  lecple 16156  occoc 16157  Atomscatm 35072  LHypclh 35793  LTrncltrn 35910  TEndoctendo 36562  DIsoCcdic 36982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-1st 7319  df-2nd 7320  df-dic 36983
This theorem is referenced by:  dicelval2N  36992
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