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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemyeb | Structured version Visualization version GIF version |
Description: Lemma for dath 36424. Frequently-used utility lemma. (Contributed by NM, 13-Aug-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))))) |
dalemyeb.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
dalemyeb | ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . 3 ⊢ (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 | dalemyeo 36320 | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
3 | eqid 2797 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dalemyeb.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
5 | 3, 4 | lplnbase 36222 | . 2 ⊢ (𝑌 ∈ 𝑂 → 𝑌 ∈ (Base‘𝐾)) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 Basecbs 16316 HLchlt 36038 LPlanesclpl 36180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-iota 6196 df-fun 6234 df-fv 6240 df-lplanes 36187 |
This theorem is referenced by: dalem5 36355 dalem8 36358 dalem-cly 36359 dalem10 36361 dalem13 36364 dalem16 36367 dalem17 36368 dalem19 36370 dalem21 36382 dalem24 36385 dalem25 36386 dalem39 36399 dalem54 36414 dalem55 36415 |
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