Theorem cdlemkuvN 38002

Description: Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma₁ (p) function 𝑈. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemk1.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk1.l	⊢ ≤ = (le‘𝐾)
cdlemk1.j	⊢ ∨ = (join‘𝐾)
cdlemk1.m	⊢ ∧ = (meet‘𝐾)
cdlemk1.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk1.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk1.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk1.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ◡𝐹))))))
cdlemk1.o	⊢ 𝑂 = (𝑆‘𝐷)
cdlemk1.u	⊢ 𝑈 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))

Assertion

Ref	Expression
cdlemkuvN	⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))

Distinct variable groups:   𝑓,𝑖, ∧   ≤ ,𝑖   ∨ ,𝑓,𝑖   𝐴,𝑖   𝐷,𝑓,𝑖   𝑓,𝐹,𝑖   𝑖,𝐻   𝑖,𝐾   𝑓,𝑁,𝑖   𝑃,𝑓,𝑖   𝑅,𝑓,𝑖   𝑇,𝑓,𝑖   𝑓,𝑊,𝑖   ∧ ,𝑒   ∨ ,𝑒   𝐷,𝑒   𝑒,𝑗,𝐺   𝑒,𝑂   𝑃,𝑒   𝑅,𝑒   𝑇,𝑒   𝑒,𝑊

Allowed substitution hints:   𝐴(𝑒,𝑓,𝑗)   𝐵(𝑒,𝑓,𝑖,𝑗)   𝐷(𝑗)   𝑃(𝑗)   𝑅(𝑗)   𝑆(𝑒,𝑓,𝑖,𝑗)   𝑇(𝑗)   𝑈(𝑒,𝑓,𝑖,𝑗)   𝐹(𝑒,𝑗)   𝐺(𝑓,𝑖)   𝐻(𝑒,𝑓,𝑗)   ∨ (𝑗)   𝐾(𝑒,𝑓,𝑗)   ≤ (𝑒,𝑓,𝑗)   ∧ (𝑗)   𝑁(𝑒,𝑗)   𝑂(𝑓,𝑖,𝑗)   𝑊(𝑗)

Proof of Theorem cdlemkuvN

Step	Hyp	Ref	Expression
1		cdlemk1.b	. 2 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemk1.l	. 2 ⊢ ≤ = (le‘𝐾)
3		cdlemk1.j	. 2 ⊢ ∨ = (join‘𝐾)
4		cdlemk1.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdlemk1.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemk1.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7		cdlemk1.r	. 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
8		cdlemk1.m	. 2 ⊢ ∧ = (meet‘𝐾)
9		cdlemk1.u	. 2 ⊢ 𝑈 = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝑒 ∘ ◡𝐷))))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdlemksv 37982	1 ⊢ (𝐺 ∈ 𝑇 → (𝑈‘𝐺) = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐷))))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ↦ cmpt 5148  ^◡ccnv 5556   ∘ ccom 5561  ‘cfv 6357  ℩crio 7115  (class class class)co 7158  Basecbs 16485  lecple 16574  joincjn 17556  meetcmee 17557  Atomscatm 36401  LHypclh 37122  LTrncltrn 37239  trLctrl 37296

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-riota 7116  df-ov 7161

This theorem is referenced by: (None)