cdleme29c - Mathbox for Norm Megill

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Theorem cdleme29c 41542

Description: Transform cdleme28b 41537. (Compare cdleme25c 41521.) TODO: FIX COMMENT. (Contributed by NM, 8-Feb-2013.)

**Hypotheses**
Ref	Expression
cdleme26.b	⊢ 𝐵 = (Base‘𝐾)
cdleme26.l	⊢ ≤ = (le‘𝐾)
cdleme26.j	⊢ ∨ = (join‘𝐾)
cdleme26.m	⊢ ∧ = (meet‘𝐾)
cdleme26.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme26.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme27.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme27.f	⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme27.z	⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme27.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
cdleme27.d	⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
cdleme27.c	⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)

**Assertion**
Ref	Expression
cdleme29c	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))

Distinct variable groups: 𝑢,𝑠,𝑧,𝐴 𝐵,𝑠,𝑢,𝑧 𝑢,𝐹 𝐻,𝑠,𝑧 ∨ ,𝑠,𝑢,𝑧 𝐾,𝑠,𝑧 ≤ ,𝑠,𝑢,𝑧 ∧ ,𝑠,𝑢,𝑧 𝑢,𝑁 𝑃,𝑠,𝑢,𝑧 𝑄,𝑠,𝑢,𝑧 𝑈,𝑠,𝑢,𝑧 𝑊,𝑠,𝑢,𝑧 𝑋,𝑠 𝑣,𝐴 𝑣,𝐵 𝑣, ∨ 𝑣, ≤ 𝑣, ∧ 𝑣,𝑃 𝑣,𝑄 𝑣,𝑈 𝑣,𝑊 𝑣,𝐶 𝑣,𝑠,𝑍,𝑢 𝑧,𝑣,𝑋 Allowed substitution hints: 𝐶(𝑧, 𝑢, 𝑠) 𝐷(𝑧, 𝑣, 𝑢, 𝑠) 𝐹(𝑧, 𝑣, 𝑠) 𝐻(𝑣, 𝑢) 𝐾(𝑣, 𝑢) 𝑁(𝑧, 𝑣, 𝑠) 𝑋(𝑢) 𝑍(𝑧)

**Proof of Theorem cdleme29c**
Step	Hyp	Ref	Expression
1		cdleme26.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdleme26.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdleme26.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdleme26.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdleme26.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdleme26.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdleme27.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdleme27.f	. . 3 ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
9		cdleme27.z	. . 3 ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
10		cdleme27.n	. . 3 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
11		cdleme27.d	. . 3 ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
12		cdleme27.c	. . 3 ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdleme29b 41541	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))
14		simp11 1222	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3 1156	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
16	1, 2, 3, 4, 5, 6	lhpmcvr2 41190	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
17	14, 15, 16	syl2anc 596	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
18		reusv1 5359	. . 3 ⊢ (∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → (∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))) ↔ ∃𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊)))))
19	17, 18	syl 18	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))) ↔ ∃𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊)))))
20	13, 19	mpbird 260	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄 ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃!𝑣 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑣 = (𝐶 ∨ (𝑋 ∧ 𝑊))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ifcif 4481 class class class wbr 5102 ‘cfv 6543 ℩crio 7380 (class class class)co 7424 Basecbs 17396 lecple 17444 joincjn 18494 meetcmee 18495 Atomscatm 40429 HLchlt 40516 LHypclh 41150

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-riotaBAD 40119

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-undef 8297 df-proset 18477 df-poset 18496 df-plt 18511 df-lub 18527 df-glb 18528 df-join 18529 df-meet 18530 df-p0 18606 df-p1 18607 df-lat 18615 df-clat 18682 df-oposet 40342 df-ol 40344 df-oml 40345 df-covers 40432 df-ats 40433 df-atl 40464 df-cvlat 40488 df-hlat 40517 df-llines 40664 df-lplanes 40665 df-lvols 40666 df-lines 40667 df-psubsp 40669 df-pmap 40670 df-padd 40962 df-lhyp 41154

This theorem is used by: cdleme29cl 41543

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