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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk1 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.) |
Ref | Expression |
---|---|
cdlemk.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk.l | ⊢ ≤ = (le‘𝐾) |
cdlemk.j | ⊢ ∨ = (join‘𝐾) |
cdlemk.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemk1 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑁‘𝑃)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3l 1203 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑅‘𝐹) = (𝑅‘𝑁)) | |
2 | 1 | oveq2d 7229 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝑅‘𝑁))) |
3 | simp1 1138 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | simp2l 1201 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝐹 ∈ 𝑇) | |
5 | simp3r 1204 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
6 | cdlemk.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
7 | cdlemk.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
8 | cdlemk.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | cdlemk.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
10 | cdlemk.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | cdlemk.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
12 | 6, 7, 8, 9, 10, 11 | trljat3 37919 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
13 | 3, 4, 5, 12 | syl3anc 1373 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
14 | simp2r 1202 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → 𝑁 ∈ 𝑇) | |
15 | 6, 7, 8, 9, 10, 11 | trljat1 37917 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃))) |
16 | 3, 14, 5, 15 | syl3anc 1373 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑅‘𝑁)) = (𝑃 ∨ (𝑁‘𝑃))) |
17 | 2, 13, 16 | 3eqtr3rd 2786 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((𝑅‘𝐹) = (𝑅‘𝑁) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))) → (𝑃 ∨ (𝑁‘𝑃)) = ((𝐹‘𝑃) ∨ (𝑅‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lecple 16809 joincjn 17818 Atomscatm 37014 HLchlt 37101 LHypclh 37735 LTrncltrn 37852 trLctrl 37909 |
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 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
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 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-map 8510 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 |
This theorem is referenced by: cdlemk5 38587 |
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