Theorem cdlemkuv-2N 40872

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma₂ (p) function, given 𝑉. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemk2.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk2.l	⊢ ≤ = (le‘𝐾)
cdlemk2.j	⊢ ∨ = (join‘𝐾)
cdlemk2.m	⊢ ∧ = (meet‘𝐾)
cdlemk2.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk2.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk2.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk2.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))))
cdlemk2.q	⊢ 𝑄 = (𝑆‘𝐶)
cdlemk2.v	⊢ 𝑉 = (𝑑 ∈ 𝑇 ↦ (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝑑)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑑 ∘ ◡𝐶))))))

Assertion

Ref	Expression
cdlemkuv-2N	⊢ (𝐺 ∈ 𝑇 → (𝑉‘𝐺) = (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐶))))))

Distinct variable groups:   𝑓,𝑖, ∧   ≤ ,𝑖   ∨ ,𝑓,𝑖   𝐴,𝑖   𝐶,𝑓,𝑖   𝑓,𝐹,𝑖   𝑖,𝐻   𝑖,𝐾   𝑓,𝑁,𝑖   𝑃,𝑓,𝑖   𝑅,𝑓,𝑖   𝑇,𝑓,𝑖   𝑓,𝑊,𝑖   ∧ ,𝑑   ∨ ,𝑑   𝐶,𝑑   𝑘,𝑑,𝐺   𝑄,𝑑   𝑃,𝑑   𝑅,𝑑   𝑇,𝑑   𝑊,𝑑

Allowed substitution hints:   𝐴(𝑓,𝑘,𝑑)   𝐵(𝑓,𝑖,𝑘,𝑑)   𝐶(𝑘)   𝑃(𝑘)   𝑄(𝑓,𝑖,𝑘)   𝑅(𝑘)   𝑆(𝑓,𝑖,𝑘,𝑑)   𝑇(𝑘)   𝐹(𝑘,𝑑)   𝐺(𝑓,𝑖)   𝐻(𝑓,𝑘,𝑑)   ∨ (𝑘)   𝐾(𝑓,𝑘,𝑑)   ≤ (𝑓,𝑘,𝑑)   ∧ (𝑘)   𝑁(𝑘,𝑑)   𝑉(𝑓,𝑖,𝑘,𝑑)   𝑊(𝑘)

Proof of Theorem cdlemkuv-2N

Step	Hyp	Ref	Expression
1		cdlemk2.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemk2.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemk2.j	. 2 ⊢ ∨ = (join‘𝐾)
4		cdlemk2.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdlemk2.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemk2.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		cdlemk2.r	. 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
8		cdlemk2.m	. 2 ⊢ ∧ = (meet‘𝐾)
9		cdlemk2.v	. 2 ⊢ 𝑉 = (𝑑 ∈ 𝑇 ↦ (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝑑)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑑 ∘ ◡𝐶))))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdlemksv 40833	1 ⊢ (𝐺 ∈ 𝑇 → (𝑉‘𝐺) = (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐶))))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5190  ^◡ccnv 5639   ∘ ccom 5644  ‘cfv 6513  ℩crio 7345  (class class class)co 7389  Basecbs 17185  lecple 17233  joincjn 18278  meetcmee 18279  Atomscatm 39251  LHypclh 39973  LTrncltrn 40090  trLctrl 40147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-riota 7346  df-ov 7392

This theorem is referenced by: (None)