Theorem dicelval1sta 41226

Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.)

Hypotheses

Ref	Expression
dicelval1sta.l	⊢ ≤ = (le‘𝐾)
dicelval1sta.a	⊢ 𝐴 = (Atoms‘𝐾)
dicelval1sta.h	⊢ 𝐻 = (LHyp‘𝐾)
dicelval1sta.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dicelval1sta.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicelval1sta.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

Assertion

Ref	Expression
dicelval1sta	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))

Distinct variable groups:   𝑔,𝐾   𝑄,𝑔   𝑇,𝑔   𝑔,𝑊

Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≤ (𝑔)   𝑉(𝑔)   𝑌(𝑔)

Proof of Theorem dicelval1sta

Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicelval1sta.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
2		dicelval1sta.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
3		dicelval1sta.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
4		dicelval1sta.p	. . . . . 6 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		dicelval1sta.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		eqid 2731	. . . . . 6 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
7		dicelval1sta.i	. . . . . 6 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dicval 41215	. . . . 5 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
9	8	eleq2d 2817	. . . 4 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))}))
10	9	biimp3a 1471	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))})
11		eqeq1 2735	. . . . 5 ⊢ (𝑓 = (1st ‘𝑌) → (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ (1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
12	11	anbi1d 631	. . . 4 ⊢ (𝑓 = (1st ‘𝑌) → ((𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
13		fveq1 6816	. . . . . 6 ⊢ (𝑠 = (2nd ‘𝑌) → (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
14	13	eqeq2d 2742	. . . . 5 ⊢ (𝑠 = (2nd ‘𝑌) → ((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
15		eleq1 2819	. . . . 5 ⊢ (𝑠 = (2nd ‘𝑌) → (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ↔ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
16	14, 15	anbi12d 632	. . . 4 ⊢ (𝑠 = (2nd ‘𝑌) → (((1st ‘𝑌) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊))))
17	12, 16	elopabi 7989	. . 3 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))} → ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
18	10, 17	syl 17	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ ((TEndo‘𝐾)‘𝑊)))
19	18	simpld 494	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑄)) → (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   class class class wbr 5086  {copab 5148  ‘cfv 6476  ℩crio 7297  1^st c1st 7914  2^nd c2nd 7915  lecple 17163  occoc 17164  Atomscatm 39302  LHypclh 40023  LTrncltrn 40140  TEndoctendo 40791  DIsoCcdic 41211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-1st 7916  df-2nd 7917  df-dic 41212

This theorem is referenced by:  dicvaddcl  41229  dicvscacl  41230