cdlemeg47b - Mathbox for Norm Megill

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Theorem cdlemeg47b 40457

Description: TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013.)

**Hypotheses**
Ref	Expression
cdlemef47.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef47.l	⊢ ≤ = (le‘𝐾)
cdlemef47.j	⊢ ∨ = (join‘𝐾)
cdlemef47.m	⊢ ∧ = (meet‘𝐾)
cdlemef47.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef47.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef47.v	⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdlemef47.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdlemefs47.o	⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdlemef47.g	⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))

**Assertion**
Ref	Expression
cdlemeg47b	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐺‘𝑆) = ⦋𝑆 / 𝑣⦌𝑁)

Distinct variable groups: 𝑎,𝑏,𝑐,𝑢,𝑣,𝐴 𝐵,𝑎,𝑏,𝑐,𝑢,𝑣 𝐻,𝑎,𝑏,𝑐,𝑢,𝑣 ∨ ,𝑎,𝑏,𝑐,𝑢,𝑣 𝐾,𝑎,𝑏,𝑐,𝑢,𝑣 ≤ ,𝑎,𝑏,𝑐,𝑢,𝑣 ∧ ,𝑎,𝑏,𝑐,𝑢,𝑣 𝑁,𝑎,𝑏,𝑐,𝑢 𝑂,𝑎,𝑏,𝑐 𝑃,𝑎,𝑏,𝑐,𝑢,𝑣 𝑄,𝑎,𝑏,𝑐,𝑢,𝑣 𝑆,𝑎,𝑏,𝑐,𝑢,𝑣 𝑉,𝑎,𝑏,𝑐,𝑢,𝑣 𝑊,𝑎,𝑏,𝑐,𝑢,𝑣 Allowed substitution hints: 𝐺(𝑣,𝑢,𝑎,𝑏,𝑐) 𝑁(𝑣) 𝑂(𝑣,𝑢)

**Proof of Theorem cdlemeg47b**
Step	Hyp	Ref	Expression
1		cdlemef47.j	. . 3 ⊢ ∨ = (join‘𝐾)
2		cdlemef47.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	cdleme46f2g2 40442	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ≠ 𝑃 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑃)))
4		cdlemef47.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
5		cdlemef47.l	. . 3 ⊢ ≤ = (le‘𝐾)
6		cdlemef47.m	. . 3 ⊢ ∧ = (meet‘𝐾)
7		cdlemef47.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
8		cdlemef47.v	. . 3 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
9		cdlemef47.n	. . 3 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
10		cdlemef47.g	. . 3 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
11	4, 5, 1, 6, 2, 7, 8, 9, 10	cdlemefr45 40376	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ≠ 𝑃 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑃)) → (𝐺‘𝑆) = ⦋𝑆 / 𝑣⦌𝑁)
12	3, 11	syl 17	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → (𝐺‘𝑆) = ⦋𝑆 / 𝑣⦌𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ⦋csb 3921 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6568 ℩crio 7398 (class class class)co 7443 Basecbs 17252 lecple 17312 joincjn 18375 meetcmee 18376 Atomscatm 39211 HLchlt 39298 LHypclh 39933

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7764

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5701 df-rel 5702 df-cnv 5703 df-co 5704 df-dm 5705 df-rn 5706 df-res 5707 df-ima 5708 df-iota 6520 df-fun 6570 df-fn 6571 df-f 6572 df-f1 6573 df-fo 6574 df-f1o 6575 df-fv 6576 df-riota 7399 df-ov 7446 df-oprab 7447 df-mpo 7448 df-1st 8024 df-2nd 8025 df-proset 18359 df-poset 18377 df-plt 18394 df-lub 18410 df-glb 18411 df-join 18412 df-meet 18413 df-p0 18489 df-p1 18490 df-lat 18496 df-clat 18563 df-oposet 39124 df-ol 39126 df-oml 39127 df-covers 39214 df-ats 39215 df-atl 39246 df-cvlat 39270 df-hlat 39299 df-lines 39450 df-psubsp 39452 df-pmap 39453 df-padd 39745 df-lhyp 39937

This theorem is referenced by: cdlemeg47rv2 40459 cdlemeg46bOLDN 40461 cdlemeg46c 40462 cdlemeg46rjgN 40471

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