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Mirrors > Home > MPE Home > Th. List > Mathboxes > trlat | Structured version Visualization version GIF version |
Description: If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.) |
Ref | Expression |
---|---|
trlat.l | ⊢ ≤ = (le‘𝐾) |
trlat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
trlat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
trlat.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
trlat.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
trlat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1134 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp3l 1199 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝐹 ∈ 𝑇) | |
3 | simp2 1135 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
4 | trlat.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
5 | eqid 2728 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | eqid 2728 | . . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾) | |
7 | trlat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | trlat.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | trlat.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | trlat.r | . . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
11 | 4, 5, 6, 7, 8, 9, 10 | trlval2 39636 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊)) |
12 | 1, 2, 3, 11 | syl3anc 1369 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) = ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊)) |
13 | simp2l 1197 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ∈ 𝐴) | |
14 | 4, 7, 8, 9 | ltrnat 39613 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴) → (𝐹‘𝑃) ∈ 𝐴) |
15 | 1, 2, 13, 14 | syl3anc 1369 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ∈ 𝐴) |
16 | simp3r 1200 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝐹‘𝑃) ≠ 𝑃) | |
17 | 16 | necomd 2993 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → 𝑃 ≠ (𝐹‘𝑃)) |
18 | 4, 5, 6, 7, 8 | lhpat 39516 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ 𝑃 ≠ (𝐹‘𝑃))) → ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊) ∈ 𝐴) |
19 | 1, 3, 15, 17, 18 | syl112anc 1372 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → ((𝑃(join‘𝐾)(𝐹‘𝑃))(meet‘𝐾)𝑊) ∈ 𝐴) |
20 | 12, 19 | eqeltrd 2829 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 lecple 17240 joincjn 18303 meetcmee 18304 Atomscatm 38735 HLchlt 38822 LHypclh 39457 LTrncltrn 39574 trLctrl 39631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 |
This theorem is referenced by: trlator0 39644 trlnidat 39646 trlnle 39659 trlval3 39660 trlval4 39661 cdlemc5 39668 cdlemg17dALTN 40137 cdlemg27a 40165 cdlemg31b0N 40167 cdlemg27b 40169 cdlemg31c 40172 cdlemg35 40186 dia2dimlem1 40537 dia2dimlem2 40538 dia2dimlem3 40539 |
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