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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem27 | Structured version Visualization version GIF version |
Description: Lemma for dath 35022. Show that the line 𝐺𝑃 intersects the dummy center of perspectivity 𝑐. (Contributed by NM, 8-Aug-2012.) |
Ref | Expression |
---|---|
dalem.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))))) |
dalem.l | ⊢ ≤ = (le‘𝐾) |
dalem.j | ⊢ ∨ = (join‘𝐾) |
dalem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem.ps | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
dalem23.m | ⊢ ∧ = (meet‘𝐾) |
dalem23.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem23.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem23.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem23.g | ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
Ref | Expression |
---|---|
dalem27 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem23.g | . . 3 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
2 | dalem.ph | . . . . . 6 ⊢ (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))))) | |
3 | 2 | dalemkelat 34910 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
4 | 3 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
5 | 2 | dalemkehl 34909 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
6 | 5 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
7 | dalem.ps | . . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
8 | 7 | dalemccea 34969 | . . . . . 6 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
9 | 8 | 3ad2ant3 1084 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) |
10 | 2 | dalempea 34912 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
11 | 10 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴) |
12 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
13 | dalem.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
14 | dalem.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 12, 13, 14 | hlatjcl 34653 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) |
16 | 6, 9, 11, 15 | syl3anc 1326 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) |
17 | 7 | dalemddea 34970 | . . . . . 6 ⊢ (𝜓 → 𝑑 ∈ 𝐴) |
18 | 17 | 3ad2ant3 1084 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴) |
19 | 2 | dalemsea 34915 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
20 | 19 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴) |
21 | 12, 13, 14 | hlatjcl 34653 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) |
22 | 6, 18, 20, 21 | syl3anc 1326 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) |
23 | dalem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
24 | dalem23.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
25 | 12, 23, 24 | latmle1 17076 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃)) |
26 | 4, 16, 22, 25 | syl3anc 1326 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃)) |
27 | 1, 26 | syl5eqbr 4688 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑐 ∨ 𝑃)) |
28 | dalem23.o | . . . 4 ⊢ 𝑂 = (LPlanes‘𝐾) | |
29 | dalem23.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
30 | dalem23.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
31 | 2, 23, 13, 14, 7, 24, 28, 29, 30, 1 | dalem23 34982 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
32 | 2, 23, 13, 14, 28, 29 | dalemply 34940 | . . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑌) |
33 | 32 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌) |
34 | 2, 23, 13, 14, 7, 24, 28, 29, 30, 1 | dalem24 34983 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌) |
35 | nbrne2 4673 | . . . . 5 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝑃 ≠ 𝐺) | |
36 | 35 | necomd 2849 | . . . 4 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝐺 ≠ 𝑃) |
37 | 33, 34, 36 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝑃) |
38 | 23, 13, 14 | hlatexch2 34682 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝐺 ≠ 𝑃) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃))) |
39 | 6, 31, 9, 11, 37, 38 | syl131anc 1339 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃))) |
40 | 27, 39 | mpd 15 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 joincjn 16944 meetcmee 16945 Latclat 17045 Atomscatm 34550 HLchlt 34637 LPlanesclpl 34778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 |
This theorem is referenced by: dalem28 34986 dalem32 34990 dalem51 35009 dalem52 35010 |
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