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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2llnneN | Structured version Visualization version GIF version |
Description: Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2lnne.l | ⊢ ≤ = (le‘𝐾) |
2lnne.j | ⊢ ∨ = (join‘𝐾) |
2lnne.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
2llnneN | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑃) ≠ (𝑅 ∨ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL) | |
2 | simp21 1208 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴) | |
3 | simp23 1210 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝑅 ∈ 𝐴) | |
4 | simp21 1208 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴) | |
5 | simp23 1210 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ 𝐴) | |
6 | simp22 1209 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴) | |
7 | 4, 5, 6 | 3jca 1130 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) |
8 | 2lnne.l | . . . . . . . 8 ⊢ ≤ = (le‘𝐾) | |
9 | 2lnne.j | . . . . . . . 8 ⊢ ∨ = (join‘𝐾) | |
10 | 2lnne.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | 8, 9, 10 | hlatexch2 37145 | . . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑅 ≤ (𝑃 ∨ 𝑄))) |
12 | 7, 11 | syld3an2 1413 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ≤ (𝑅 ∨ 𝑄) → 𝑅 ≤ (𝑃 ∨ 𝑄))) |
13 | 12 | con3d 155 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑃 ≤ (𝑅 ∨ 𝑄))) |
14 | 13 | 3exp 1121 | . . . 4 ⊢ (𝐾 ∈ HL → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ≠ 𝑄 → (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑃 ≤ (𝑅 ∨ 𝑄))))) |
15 | 14 | imp4a 426 | . . 3 ⊢ (𝐾 ∈ HL → ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑃 ≤ (𝑅 ∨ 𝑄)))) |
16 | 15 | 3imp 1113 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑅 ∨ 𝑄)) |
17 | 8, 9, 10 | 2llnne2N 37157 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑃 ≤ (𝑅 ∨ 𝑄)) → (𝑅 ∨ 𝑃) ≠ (𝑅 ∨ 𝑄)) |
18 | 1, 2, 3, 16, 17 | syl121anc 1377 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ∨ 𝑃) ≠ (𝑅 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2939 class class class wbr 5050 ‘cfv 6377 (class class class)co 7210 lecple 16806 joincjn 17815 Atomscatm 37012 HLchlt 37099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-proset 17799 df-poset 17817 df-plt 17833 df-lub 17849 df-glb 17850 df-join 17851 df-meet 17852 df-p0 17928 df-lat 17935 df-covers 37015 df-ats 37016 df-atl 37047 df-cvlat 37071 df-hlat 37100 |
This theorem is referenced by: (None) |
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