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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem18 | Structured version Visualization version GIF version |
Description: Lemma for dath 38410. Show that a dummy atom 𝑐 exists outside of the 𝑌 and 𝑍 planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-Jul-2012.) |
Ref | Expression |
---|---|
dalema.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem18.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
Ref | Expression |
---|---|
dalem18 | ⊢ (𝜑 → ∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) | |
2 | 1 | dalemkehl 38297 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | dalempea 38300 | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
4 | 1 | dalemqea 38301 | . . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
5 | 1 | dalemrea 38302 | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝐴) |
6 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
7 | dalemc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
8 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | 6, 7, 8 | 3dim3 38143 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
10 | 2, 3, 4, 5, 9 | syl13anc 1372 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
11 | dalem18.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
12 | 11 | breq2i 5149 | . . . 4 ⊢ (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
13 | 12 | notbii 319 | . . 3 ⊢ (¬ 𝑐 ≤ 𝑌 ↔ ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
14 | 13 | rexbii 3093 | . 2 ⊢ (∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ↔ ∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
15 | 10, 14 | sylibr 233 | 1 ⊢ (𝜑 → ∃𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 lecple 17186 joincjn 18246 Atomscatm 37936 HLchlt 38023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 |
This theorem is referenced by: dalem20 38367 |
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