Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg4b2 | Structured version Visualization version GIF version |
Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg4.l | ⊢ ≤ = (le‘𝐾) |
cdlemg4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg4.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg4.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg4.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemg4.j | ⊢ ∨ = (join‘𝐾) |
cdlemg4b.v | ⊢ 𝑉 = (𝑅‘𝐺) |
Ref | Expression |
---|---|
cdlemg4b2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∨ 𝑉) = (𝑃 ∨ (𝐺‘𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg4b.v | . . . 4 ⊢ 𝑉 = (𝑅‘𝐺) | |
2 | cdlemg4.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg4.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
4 | eqid 2610 | . . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
5 | cdlemg4.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg4.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg4.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | cdlemg4.r | . . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
9 | 2, 3, 4, 5, 6, 7, 8 | trlval2 34468 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
10 | 9 | 3com23 1263 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → (𝑅‘𝐺) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
11 | 1, 10 | syl5eq 2656 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → 𝑉 = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) |
12 | 11 | oveq2d 6565 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∨ 𝑉) = ((𝐺‘𝑃) ∨ ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊))) |
13 | simp1 1054 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | simp2l 1080 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → 𝑃 ∈ 𝐴) | |
15 | 2, 5, 6, 7 | ltrnel 34443 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) |
16 | 15 | 3com23 1263 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊)) |
17 | eqid 2610 | . . . 4 ⊢ ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊) = ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊) | |
18 | 2, 3, 4, 5, 6, 17 | cdleme0cq 34520 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ((𝐺‘𝑃) ∈ 𝐴 ∧ ¬ (𝐺‘𝑃) ≤ 𝑊))) → ((𝐺‘𝑃) ∨ ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) = (𝑃 ∨ (𝐺‘𝑃))) |
19 | 13, 14, 16, 18 | syl12anc 1316 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∨ ((𝑃 ∨ (𝐺‘𝑃))(meet‘𝐾)𝑊)) = (𝑃 ∨ (𝐺‘𝑃))) |
20 | 12, 19 | eqtrd 2644 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐺 ∈ 𝑇) → ((𝐺‘𝑃) ∨ 𝑉) = (𝑃 ∨ (𝐺‘𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 |
This theorem is referenced by: cdlemg4b12 34917 cdlemg4c 34918 |
Copyright terms: Public domain | W3C validator |