cdleme29c - Mathbox for Norm Megill

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Theorem cdleme29c 38317

Description: Transform cdleme28b 38312. (Compare cdleme25c 38296.) 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 38316	. 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 1201	. . . 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 1136	. . . 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 37965	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑠 ∈ 𝐴 (¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
17	14, 15, 16	syl2anc 583	. . 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 5315	. . 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 17	. 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 256	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ∃!wreu 3065 ifcif 4456 class class class wbr 5070 ‘cfv 6418 ℩crio 7211 (class class class)co 7255 Basecbs 16840 lecple 16895 joincjn 17944 meetcmee 17945 Atomscatm 37204 HLchlt 37291 LHypclh 37925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-undef 8060 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929

This theorem is referenced by: cdleme29cl 38318

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