Theorem dicval2 42502

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 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)

Assertion

Ref	Expression
dicval2	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})

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

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	dicval 42499	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
9		dicval2.g	. . . . . 6 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
10	9	fveq2i 6919	. . . . 5 ⊢ (𝑠‘𝐺) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
11	10	eqeq2i 2776	. . . 4 ⊢ (𝑓 = (𝑠‘𝐺) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
12	11	anbi1i 636	. . 3 ⊢ ((𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸))
13	12	opabbii 5178	. 2 ⊢ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)} = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)}
14	8, 13	eqtr4di 2816	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘𝐺) ∧ 𝑠 ∈ 𝐸)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   class class class wbr 5108  {copab 5173  ‘cfv 6566  ℩crio 7407  lecple 17479  occoc 17480  Atomscatm 40586  LHypclh 41307  LTrncltrn 41424  TEndoctendo 42075  DIsoCcdic 42495

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-dic 42496

This theorem is used by:  dicelval3  42503  diclspsn  42517  dih1dimatlem  42652