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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemv | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 36096. (Contributed by NM, 23-Nov-2012.) |
Ref | Expression |
---|---|
4thatlem.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
4thatlem0.l | ⊢ ≤ = (le‘𝐾) |
4thatlem0.j | ⊢ ∨ = (join‘𝐾) |
4thatlem0.m | ⊢ ∧ = (meet‘𝐾) |
4thatlem0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
4thatlem0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
4thatlem0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
4thatlem0.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
4atexlemv | ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4thatlem.ph | . . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | |
2 | 1 | 4atexlemk 36068 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | 4atexlemw 36069 | . 2 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
4 | 1 | 4atexlempw 36070 | . 2 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
5 | 1 | 4atexlems 36073 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
6 | 4thatlem0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | 4thatlem0.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
8 | 4thatlem0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 1, 6, 7, 8 | 4atexlempns 36083 | . 2 ⊢ (𝜑 → 𝑃 ≠ 𝑆) |
10 | 4thatlem0.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
11 | 4thatlem0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
12 | 4thatlem0.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
13 | 6, 7, 10, 8, 11, 12 | lhpat2 36066 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → 𝑉 ∈ 𝐴) |
14 | 2, 3, 4, 5, 9, 13 | syl212anc 1500 | 1 ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 lecple 16274 joincjn 17259 meetcmee 17260 Atomscatm 35284 HLchlt 35371 LHypclh 36005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-lhyp 36009 |
This theorem is referenced by: 4atexlemunv 36087 4atexlemtlw 36088 4atexlemntlpq 36089 4atexlemc 36090 4atexlemnclw 36091 |
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