Theorem dicelval2N 41956

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 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)

Assertion

Ref	Expression
dicelval2N	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 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

Step	Hyp	Ref	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)
8	1, 2, 3, 4, 5, 6, 7	dicelvalN 41952	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))))
9		dicval2.g	. . . . . 6 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
10	9	fveq2i 6884	. . . . 5 ⊢ ((2nd ‘𝑌)‘𝐺) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
11	10	eqeq2i 2776	. . . 4 ⊢ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ↔ (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
12	11	anbi1i 635	. . 3 ⊢ (((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸) ↔ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))
13	12	anbi2i 634	. 2 ⊢ ((𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸)) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸)))
14	8, 13	bitr4di 292	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   class class class wbr 5109   × cxp 5659  ‘cfv 6536  ℩crio 7366  1^st c1st 7980  2^nd c2nd 7981  lecple 17312  occoc 17313  Atomscatm 40037  LHypclh 40758  LTrncltrn 40875  TEndoctendo 41526  DIsoCcdic 41946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-1st 7982  df-2nd 7983  df-dic 41947

This theorem is referenced by: (None)