cdlemk18-3N - Mathbox for Norm Megill

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Theorem cdlemk18-3N 40861

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. 𝑁, 𝑌, 𝑂, 𝐷 are k, sigma₂ (p), k₁, f₁. (Contributed by NM, 7-Jul-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
cdlemk3.b	⊢ 𝐵 = (Base‘𝐾)
cdlemk3.l	⊢ ≤ = (le‘𝐾)
cdlemk3.j	⊢ ∨ = (join‘𝐾)
cdlemk3.m	⊢ ∧ = (meet‘𝐾)
cdlemk3.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemk3.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemk3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemk3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemk3.s	⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
cdlemk3.u1	⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑))))))

**Assertion**
Ref	Expression
cdlemk18-3N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐹)‘𝑃) = (𝑁‘𝑃))

Distinct variable groups: 𝑒,𝑑,𝑓,𝑖, ∧ ≤ ,𝑖 ∨ ,𝑑,𝑒,𝑓,𝑖 𝐴,𝑖 𝑗,𝑑,𝐷,𝑒,𝑓,𝑖 𝑓,𝐹,𝑖 𝑖,𝐻 𝑖,𝐾 𝑓,𝑁,𝑖 𝑃,𝑑,𝑒,𝑓,𝑖 𝑅,𝑑,𝑒,𝑓,𝑖 𝑇,𝑑,𝑒,𝑓,𝑖 𝑊,𝑑,𝑒,𝑓,𝑖 ∧ ,𝑗 ≤ ,𝑗 ∨ ,𝑗 𝐴,𝑗 𝑗,𝐹 𝑗,𝐻 𝑗,𝐾 𝑗,𝑁 𝑃,𝑗 𝑅,𝑗 𝑆,𝑑,𝑒,𝑗 𝑇,𝑗 𝑗,𝑊 𝐹,𝑑,𝑒 Allowed substitution hints: 𝐴(𝑒,𝑓,𝑑) 𝐵(𝑒,𝑓,𝑖,𝑗,𝑑) 𝑆(𝑓,𝑖) 𝐻(𝑒,𝑓,𝑑) 𝐾(𝑒,𝑓,𝑑) ≤ (𝑒,𝑓,𝑑) 𝑁(𝑒,𝑑) 𝑌(𝑒,𝑓,𝑖,𝑗,𝑑)

**Proof of Theorem cdlemk18-3N**
Step	Hyp	Ref	Expression
1		simp22 1207	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐷 ∈ 𝑇)
2		simp21 1206	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐹 ∈ 𝑇)
3		cdlemk3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4		cdlemk3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		cdlemk3.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
6		cdlemk3.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
7		cdlemk3.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8		cdlemk3.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		cdlemk3.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10		cdlemk3.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
11		cdlemk3.s	. . . . 5 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
12		cdlemk3.u1	. . . . 5 ⊢ 𝑌 = (𝑑 ∈ 𝑇, 𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝑑)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝑑))))))
13		eqid 2734	. . . . 5 ⊢ (𝑆‘𝐷) = (𝑆‘𝐷)
14		eqid 2734	. . . . 5 ⊢ (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷)))))) = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷))))))
15	3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdlemkuu 40856	. . . 4 ⊢ ((𝐷 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) → (𝐷𝑌𝐹) = ((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷))))))‘𝐹))
16	1, 2, 15	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐷𝑌𝐹) = ((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷))))))‘𝐹))
17	16	fveq1d 6888	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐹)‘𝑃) = (((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷))))))‘𝐹)‘𝑃))
18	3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14	cdlemk18-2N 40847	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑁‘𝑃) = (((𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (𝑗‘𝑃) = ((𝑃 ∨ (𝑅‘𝑒)) ∧ (((𝑆‘𝐷)‘𝑃) ∨ (𝑅‘(𝑒 ∘ ^◡𝐷))))))‘𝐹)‘𝑃))
19	17, 18	eqtr4d 2772	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐷) ≠ (𝑅‘𝐹) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → ((𝐷𝑌𝐹)‘𝑃) = (𝑁‘𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 class class class wbr 5123 ↦ cmpt 5205 I cid 5557 ^◡ccnv 5664 ↾ cres 5667 ∘ ccom 5669 ‘cfv 6541 ℩crio 7369 (class class class)co 7413 ∈ cmpo 7415 Basecbs 17229 lecple 17280 joincjn 18327 meetcmee 18328 Atomscatm 39223 HLchlt 39310 LHypclh 39945 LTrncltrn 40062 trLctrl 40119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-riotaBAD 38913

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7996 df-2nd 7997 df-undef 8280 df-map 8850 df-proset 18310 df-poset 18329 df-plt 18344 df-lub 18360 df-glb 18361 df-join 18362 df-meet 18363 df-p0 18439 df-p1 18440 df-lat 18446 df-clat 18513 df-oposet 39136 df-ol 39138 df-oml 39139 df-covers 39226 df-ats 39227 df-atl 39258 df-cvlat 39282 df-hlat 39311 df-llines 39459 df-lplanes 39460 df-lvols 39461 df-lines 39462 df-psubsp 39464 df-pmap 39465 df-padd 39757 df-lhyp 39949 df-laut 39950 df-ldil 40065 df-ltrn 40066 df-trl 40120

This theorem is referenced by: (None)

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