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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme32sn1awN | Structured version Visualization version GIF version |
Description: Show that ⦋𝑅 / 𝑠⦌𝑁 is an atom not under 𝑊 when 𝑅 ≤ (𝑃 ∨ 𝑄). (Contributed by NM, 6-Mar-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cdleme32.b | ⊢ 𝐵 = (Base‘𝐾) |
cdleme32.l | ⊢ ≤ = (le‘𝐾) |
cdleme32.j | ⊢ ∨ = (join‘𝐾) |
cdleme32.m | ⊢ ∧ = (meet‘𝐾) |
cdleme32.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdleme32.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdleme32.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdleme32.c | ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) |
cdleme32.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdleme32.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdleme32.i | ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)) |
cdleme32.n | ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶) |
cdleme32a1.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
cdleme32a1.z | ⊢ 𝑍 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)) |
Ref | Expression |
---|---|
cdleme32sn1awN | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐴 ∧ ¬ ⦋𝑅 / 𝑠⦌𝑁 ≤ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme32.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdleme32.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdleme32.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdleme32.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdleme32.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdleme32.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdleme32.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | cdleme32.d | . 2 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdleme32.e | . 2 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdleme32.i | . 2 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)) | |
11 | cdleme32.n | . 2 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶) | |
12 | cdleme32a1.y | . 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) | |
13 | cdleme32a1.z | . 2 ⊢ 𝑍 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | cdlemefs32sn1aw 38192 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (⦋𝑅 / 𝑠⦌𝑁 ∈ 𝐴 ∧ ¬ ⦋𝑅 / 𝑠⦌𝑁 ≤ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ∀wral 3062 ⦋csb 3826 ifcif 4454 class class class wbr 5068 ‘cfv 6398 ℩crio 7188 (class class class)co 7232 Basecbs 16785 lecple 16834 joincjn 17843 meetcmee 17844 Atomscatm 37041 HLchlt 37128 LHypclh 37762 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-riotaBAD 36731 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-iin 4922 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-1st 7780 df-2nd 7781 df-undef 8036 df-proset 17827 df-poset 17845 df-plt 17861 df-lub 17877 df-glb 17878 df-join 17879 df-meet 17880 df-p0 17956 df-p1 17957 df-lat 17963 df-clat 18030 df-oposet 36954 df-ol 36956 df-oml 36957 df-covers 37044 df-ats 37045 df-atl 37076 df-cvlat 37100 df-hlat 37129 df-llines 37276 df-lplanes 37277 df-lvols 37278 df-lines 37279 df-psubsp 37281 df-pmap 37282 df-padd 37574 df-lhyp 37766 |
This theorem is referenced by: (None) |
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