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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme25cl | Structured version Visualization version GIF version |
Description: Show closure of the unique element in cdleme25c 39530. (Contributed by NM, 2-Feb-2013.) |
Ref | Expression |
---|---|
cdleme24.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme24.l | ⊢ ≤ = (le‘𝐾) |
cdleme24.j | ⊢ ∨ = (join‘𝐾) |
cdleme24.m | ⊢ ∧ = (meet‘𝐾) |
cdleme24.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme24.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme24.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme24.f | ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) |
cdleme24.n | ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) |
cdleme25cl.i | ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) |
Ref | Expression |
---|---|
cdleme25cl | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme25cl.i | . 2 ⊢ 𝐼 = (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) | |
2 | cdleme24.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | cdleme24.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | cdleme24.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | cdleme24.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
6 | cdleme24.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | cdleme24.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | cdleme24.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
9 | cdleme24.f | . . . 4 ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) | |
10 | cdleme24.n | . . . 4 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme25c 39530 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃!𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) |
12 | riotacl 7386 | . . 3 ⊢ (∃!𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁) → (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) ∈ 𝐵) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (℩𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) ∈ 𝐵) |
14 | 1, 13 | eqeltrid 2836 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → 𝐼 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃!wreu 3373 class class class wbr 5148 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17149 lecple 17209 joincjn 18269 meetcmee 18270 Atomscatm 38437 HLchlt 38524 LHypclh 39159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 |
This theorem is referenced by: cdleme26e 39534 cdleme26eALTN 39536 cdleme26fALTN 39537 cdleme26f 39538 cdleme26f2ALTN 39539 cdleme26f2 39540 cdleme27cl 39541 cdlemefs27cl 39588 cdlemefs32sn1aw 39589 cdleme43fsv1snlem 39595 cdleme41sn3a 39608 cdleme40m 39642 cdleme40n 39643 |
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