cdlemk13 - Mathbox for Norm Megill

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Theorem cdlemk13 38522

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. 𝑂, 𝐷 are k₁, f₁. (Contributed by NM, 1-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)

**Assertion**
Ref	Expression
cdlemk13	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑂‘𝑃) = ((𝑃 ∨ (𝑅‘𝐷)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐷 ∘ ^◡𝐹)))))

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 Allowed substitution hints: 𝐴(𝑓) 𝐵(𝑓,𝑖) 𝑆(𝑓,𝑖) 𝐻(𝑓) 𝐾(𝑓) ≤ (𝑓) 𝑂(𝑓,𝑖)

**Proof of Theorem cdlemk13**
Step	Hyp	Ref	Expression
1		cdlemk1.o	. . 3 ⊢ 𝑂 = (𝑆‘𝐷)
2	1	fveq1i 6688	. 2 ⊢ (𝑂‘𝑃) = ((𝑆‘𝐷)‘𝑃)
3		cdlemk1.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
4		cdlemk1.l	. . 3 ⊢ ≤ = (le‘𝐾)
5		cdlemk1.j	. . 3 ⊢ ∨ = (join‘𝐾)
6		cdlemk1.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
7		cdlemk1.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemk1.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
9		cdlemk1.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
10		cdlemk1.m	. . 3 ⊢ ∧ = (meet‘𝐾)
11		cdlemk1.s	. . 3 ⊢ 𝑆 = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (𝑖‘𝑃) = ((𝑃 ∨ (𝑅‘𝑓)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑓 ∘ ^◡𝐹))))))
12	3, 4, 5, 6, 7, 8, 9, 10, 11	cdlemksv2 38517	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → ((𝑆‘𝐷)‘𝑃) = ((𝑃 ∨ (𝑅‘𝐷)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐷 ∘ ^◡𝐹)))))
13	2, 12	syl5eq 2786	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐷) ≠ (𝑅‘𝐹))) → (𝑂‘𝑃) = ((𝑃 ∨ (𝑅‘𝐷)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝐷 ∘ ^◡𝐹)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ↦ cmpt 5120 I cid 5438 ^◡ccnv 5534 ↾ cres 5537 ∘ ccom 5539 ‘cfv 6350 ℩crio 7139 (class class class)co 7183 Basecbs 16599 lecple 16688 joincjn 17683 meetcmee 17684 Atomscatm 36933 HLchlt 37020 LHypclh 37654 LTrncltrn 37771 trLctrl 37828

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-riotaBAD 36623

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-1st 7727 df-2nd 7728 df-undef 7981 df-map 8452 df-proset 17667 df-poset 17685 df-plt 17697 df-lub 17713 df-glb 17714 df-join 17715 df-meet 17716 df-p0 17778 df-p1 17779 df-lat 17785 df-clat 17847 df-oposet 36846 df-ol 36848 df-oml 36849 df-covers 36936 df-ats 36937 df-atl 36968 df-cvlat 36992 df-hlat 37021 df-llines 37168 df-lplanes 37169 df-lvols 37170 df-lines 37171 df-psubsp 37173 df-pmap 37174 df-padd 37466 df-lhyp 37658 df-laut 37659 df-ldil 37774 df-ltrn 37775 df-trl 37829

This theorem is referenced by: cdlemkole 38523 cdlemk14 38524 cdlemk13-2N 38546

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