Theorem dicelval2N 39123

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 39119	. 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 6759	. . . . 5 ⊢ ((2nd ‘𝑌)‘𝐺) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
11	10	eqeq2i 2751	. . . 4 ⊢ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ↔ (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
12	11	anbi1i 623	. . 3 ⊢ (((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸) ↔ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))
13	12	anbi2i 622	. 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 288	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070   × cxp 5578  ‘cfv 6418  ℩crio 7211  1^st c1st 7802  2^nd c2nd 7803  lecple 16895  occoc 16896  Atomscatm 37204  LHypclh 37925  LTrncltrn 38042  TEndoctendo 38693  DIsoCcdic 39113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-1st 7804  df-2nd 7805  df-dic 39114

This theorem is referenced by: (None)