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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2fv | Structured version Visualization version GIF version |
Description: Value of a translation in terms of an associated atom. cdleme48fvg 34806 with simpler hypotheses. TODO: Use ltrnj 34436 to vastly simplify. (Contributed by NM, 23-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg2inv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg2inv.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg2j.l | ⊢ ≤ = (le‘𝐾) |
cdlemg2j.j | ⊢ ∨ = (join‘𝐾) |
cdlemg2j.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg2j.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg2j.b | ⊢ 𝐵 = (Base‘𝐾) |
Ref | Expression |
---|---|
cdlemg2fv | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg2j.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemg2j.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg2j.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg2j.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg2j.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg2inv.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg2inv.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | eqid 2610 | . 2 ⊢ ((𝑝 ∨ 𝑞) ∧ 𝑊) = ((𝑝 ∨ 𝑞) ∧ 𝑊) | |
9 | eqid 2610 | . 2 ⊢ ((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) = ((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) | |
10 | eqid 2610 | . 2 ⊢ ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
11 | eqid 2610 | . 2 ⊢ (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) = (𝑥 ∈ 𝐵 ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdlemg2fvlem 34900 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐹‘𝑋) = ((𝐹‘𝑃) ∨ (𝑋 ∧ 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⦋csb 3499 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 lecple 15775 joincjn 16767 meetcmee 16768 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 |
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 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 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-undef 7286 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-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 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: cdlemg2fv2 34906 cdlemg7fvbwN 34913 cdlemg7fvN 34930 |
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