cdlemefr45e - Mathbox for Norm Megill

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Theorem cdlemefr45e 39237

Description: Explicit expansion of cdlemefr45 39236. TODO: use to shorten cdlemefr45 39236 uses? TODO: FIX COMMENT. (Contributed by NM, 10-Apr-2013.)

**Hypotheses**
Ref	Expression
cdlemef45.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef45.l	⊢ ≤ = (le‘𝐾)
cdlemef45.j	⊢ ∨ = (join‘𝐾)
cdlemef45.m	⊢ ∧ = (meet‘𝐾)
cdlemef45.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef45.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef45.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef45.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemef45.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))

**Assertion**
Ref	Expression
cdlemefr45e	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘𝑅) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))

Distinct variable groups: 𝑡,𝑠,𝑥,𝑦,𝑧,𝐴 𝐵,𝑠,𝑡,𝑥,𝑦,𝑧 𝐷,𝑠,𝑥,𝑦,𝑧 𝑥,𝐸,𝑦,𝑧 𝐻,𝑠,𝑡,𝑥,𝑦,𝑧 ∨ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝐾,𝑠,𝑡,𝑥,𝑦,𝑧 ≤ ,𝑠,𝑡,𝑥,𝑦,𝑧 ∧ ,𝑠,𝑡,𝑥,𝑦,𝑧 𝑃,𝑠,𝑡,𝑥,𝑦,𝑧 𝑄,𝑠,𝑡,𝑥,𝑦,𝑧 𝑅,𝑠,𝑡,𝑥,𝑦,𝑧 𝑈,𝑠,𝑡,𝑥,𝑦,𝑧 𝑊,𝑠,𝑡,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐷(𝑡) 𝐸(𝑡,𝑠) 𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

**Proof of Theorem cdlemefr45e**
Step	Hyp	Ref	Expression
1		cdlemef45.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef45.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemef45.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemef45.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemef45.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef45.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef45.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef45.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemef45.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdlemefr45 39236	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘𝑅) = ⦋𝑅 / 𝑡⦌𝐷)
11		simp2rl 1243	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ∈ 𝐴)
12		eqid 2733	. . . 4 ⊢ ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
13	8, 12	cdleme31sc 39193	. . 3 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑡⦌𝐷 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
14	11, 13	syl 17	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑡⦌𝐷 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
15	10, 14	eqtrd 2773	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐹‘𝑅) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ⦋csb 3892 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 ℩crio 7359 (class class class)co 7404 Basecbs 17140 lecple 17200 joincjn 18260 meetcmee 18261 Atomscatm 38071 HLchlt 38158 LHypclh 38793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-oposet 37984 df-ol 37986 df-oml 37987 df-covers 38074 df-ats 38075 df-atl 38106 df-cvlat 38130 df-hlat 38159 df-lines 38310 df-psubsp 38312 df-pmap 38313 df-padd 38605 df-lhyp 38797

This theorem is referenced by: cdlemefs45eN 39240 cdleme46frvlpq 39313 cdleme50trn1 39358

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