cdlemkj - Mathbox for Norm Megill

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Theorem cdlemkj 37530

Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.)

**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)
cdlemk.z	⊢ 𝑍 = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))))

**Assertion**
Ref	Expression
cdlemkj	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑍 ∈ 𝑇)

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

**Proof of Theorem cdlemkj**
Step	Hyp	Ref	Expression
1		simp11l 1277	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐾 ∈ HL)
2		simp11r 1278	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
3		simp33 1204	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
4		cdlemk1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		cdlemk1.l	. . 3 ⊢ ≤ = (le‘𝐾)
6		cdlemk1.j	. . 3 ⊢ ∨ = (join‘𝐾)
7		cdlemk1.m	. . 3 ⊢ ∧ = (meet‘𝐾)
8		cdlemk1.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdlemk1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdlemk1.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		cdlemk1.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12		cdlemk1.s	. . 3 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
13		cdlemk1.o	. . 3 ⊢ 𝑂 = (𝑆‘𝐷)
14	4, 5, 6, 7, 8, 9, 10, 11, 12, 13	cdlemk16a 37523	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))) ∈ 𝐴 ∧ ¬ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))) ≤ 𝑊))
15		cdlemk.z	. . 3 ⊢ 𝑍 = (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))))
16	5, 8, 9, 10, 15	ltrniotacl 37246	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))) ∈ 𝐴 ∧ ¬ ((𝑃 ∨ (𝑅‘𝐺)) ∧ ((𝑂‘𝑃) ∨ (𝑅‘(𝐺 ∘ ^◡𝐷)))) ≤ 𝑊)) → 𝑍 ∈ 𝑇)
17	1, 2, 3, 14, 16	syl211anc 1369	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑍 ∈ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 class class class wbr 4962 ↦ cmpt 5041 I cid 5347 ^◡ccnv 5442 ↾ cres 5445 ∘ ccom 5447 ‘cfv 6225 ℩crio 6976 (class class class)co 7016 Basecbs 16312 lecple 16401 joincjn 17383 meetcmee 17384 Atomscatm 35930 HLchlt 36017 LHypclh 36651 LTrncltrn 36768 trLctrl 36825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-riotaBAD 35620

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-undef 7790 df-map 8258 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-oposet 35843 df-ol 35845 df-oml 35846 df-covers 35933 df-ats 35934 df-atl 35965 df-cvlat 35989 df-hlat 36018 df-llines 36165 df-lplanes 36166 df-lvols 36167 df-lines 36168 df-psubsp 36170 df-pmap 36171 df-padd 36463 df-lhyp 36655 df-laut 36656 df-ldil 36771 df-ltrn 36772 df-trl 36826

This theorem is referenced by: cdlemkuel 37532 cdlemkj-2N 37549

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