Theorem cdlemkuv-2N 42077

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 42038	1 ⊢ (𝐺 ∈ 𝑇 → (𝑉‘𝐺) = (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐶))))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ^◡ccnv 5650   ∘ ccom 5655  ‘cfv 6543  ℩crio 7380  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  meetcmee 18495  Atomscatm 40457  LHypclh 41178  LTrncltrn 41295  trLctrl 41352

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-riota 7381  df-ov 7427

This theorem is used by: (None)