cdlemkuv2-2 - Mathbox for Norm Megill

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Theorem cdlemkuv2-2 42088

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 2, where sigma₂ (p) is 𝑉, f₂ is 𝐶, and k₂ is 𝑄. (Contributed by NM, 2-Jul-2013.)

**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
cdlemkuv2-2	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐶 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐶) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐶) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐶 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑉‘𝐺)‘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 𝐺,𝑑,𝑘 𝑄,𝑑 𝑃,𝑑 𝑅,𝑑 𝑇,𝑑 𝑊,𝑑 ∧ ,𝑘 ≤ ,𝑘 ∨ ,𝑘 𝐴,𝑘 𝐶,𝑘 𝑘,𝐹 𝑘,𝐻 𝑘,𝐾 𝑘,𝑁 𝑄,𝑘 𝑃,𝑘 𝑅,𝑘 𝑇,𝑘 𝑘,𝑊 Allowed substitution hints: 𝐴(𝑓, 𝑑) 𝐵(𝑓, 𝑖, 𝑘, 𝑑) 𝑄(𝑓, 𝑖) 𝑆(𝑓, 𝑖, 𝑘, 𝑑) 𝐹(𝑑) 𝐺(𝑓, 𝑖) 𝐻(𝑓, 𝑑) 𝐾(𝑓, 𝑑) ≤ (𝑓, 𝑑) 𝑁(𝑑) 𝑉(𝑓, 𝑖, 𝑘, 𝑑)

**Proof of Theorem cdlemkuv2-2**
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.m	. 2 ⊢ ∧ = (meet‘𝐾)
5		cdlemk2.a	. 2 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemk2.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemk2.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		cdlemk2.r	. 2 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		cdlemk2.s	. 2 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
10		cdlemk2.q	. 2 ⊢ 𝑄 = (𝑆‘𝐶)
11		cdlemk2.v	. 2 ⊢ 𝑉 = (𝑑 ∈ 𝑇 ↦ (℩𝑘 ∈ 𝑇 (𝑘‘𝑃) = ((𝑃 ∨ (𝑅‘𝑑)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝑑 ∘ ^◡𝐶))))))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	cdlemkuv2 42070	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐶 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐶) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐶) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐶 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝑉‘𝐺)‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑄‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐶)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 class class class wbr 5106 ↦ cmpt 5189 I cid 5549 ^◡ccnv 5654 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6547 ℩crio 7386 (class class class)co 7430 Basecbs 17402 lecple 17450 joincjn 18500 meetcmee 18501 Atomscatm 40466 HLchlt 40553 LHypclh 41187 LTrncltrn 41304 trLctrl 41361

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 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-riotaBAD 40156

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 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-rmo 3369 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 4839 df-uni 4871 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8011 df-2nd 8012 df-undef 8304 df-map 8863 df-proset 18483 df-poset 18502 df-plt 18517 df-lub 18533 df-glb 18534 df-join 18535 df-meet 18536 df-p0 18612 df-p1 18613 df-lat 18621 df-clat 18688 df-oposet 40379 df-ol 40381 df-oml 40382 df-covers 40469 df-ats 40470 df-atl 40501 df-cvlat 40525 df-hlat 40554 df-llines 40701 df-lplanes 40702 df-lvols 40703 df-lines 40704 df-psubsp 40706 df-pmap 40707 df-padd 40999 df-lhyp 41191 df-laut 41192 df-ldil 41307 df-ltrn 41308 df-trl 41362

This theorem is used by: cdlemk22 42096

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