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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemrotps | Structured version Visualization version GIF version |
Description: Lemma for dath 35022. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 15-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)))) |
dalemrotps.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
Ref | Expression |
---|---|
dalemrotps | ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.ps | . . . . 5 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
2 | 1 | dalemccea 34969 | . . . 4 ⊢ (𝜓 → 𝑐 ∈ 𝐴) |
3 | 1 | dalemddea 34970 | . . . 4 ⊢ (𝜓 → 𝑑 ∈ 𝐴) |
4 | 2, 3 | jca 554 | . . 3 ⊢ (𝜓 → (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) |
5 | 4 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴)) |
6 | 1 | dalem-ccly 34971 | . . . 4 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌) |
7 | 6 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌) |
8 | dalem.ph | . . . . . . 7 ⊢ (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))))) | |
9 | dalem.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
10 | dalem.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | 8, 9, 10 | dalemqrprot 34934 | . . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
12 | dalemrotps.y | . . . . . 6 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
13 | 11, 12 | syl6reqr 2675 | . . . . 5 ⊢ (𝜑 → 𝑌 = ((𝑄 ∨ 𝑅) ∨ 𝑃)) |
14 | 13 | breq2d 4665 | . . . 4 ⊢ (𝜑 → (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃))) |
15 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃))) |
16 | 7, 15 | mtbid 314 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃)) |
17 | 1 | dalemccnedd 34973 | . . . . 5 ⊢ (𝜓 → 𝑐 ≠ 𝑑) |
18 | 17 | necomd 2849 | . . . 4 ⊢ (𝜓 → 𝑑 ≠ 𝑐) |
19 | 18 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑑 ≠ 𝑐) |
20 | 1 | dalem-ddly 34972 | . . . . 5 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌) |
21 | 20 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌) |
22 | 13 | breq2d 4665 | . . . . 5 ⊢ (𝜑 → (𝑑 ≤ 𝑌 ↔ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃))) |
23 | 22 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑑 ≤ 𝑌 ↔ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃))) |
24 | 21, 23 | mtbid 314 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃)) |
25 | 1 | dalemclccjdd 34974 | . . . 4 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
26 | 25 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑)) |
27 | 19, 24, 26 | 3jca 1242 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) |
28 | 5, 16, 27 | 3jca 1242 | 1 ⊢ ((𝜑 ∧ 𝜓) → ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
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 Atomscatm 34550 HLchlt 34637 |
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-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 |
This theorem is referenced by: dalem29 34987 dalem30 34988 dalem31N 34989 dalem32 34990 dalem33 34991 dalem34 34992 dalem35 34993 dalem36 34994 dalem37 34995 dalem40 34998 dalem46 35004 dalem47 35005 dalem49 35007 dalem50 35008 dalem58 35016 dalem59 35017 |
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