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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme50lebi | Structured version Visualization version GIF version |
Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-Apr-2013.) |
Ref | Expression |
---|---|
cdlemef50.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemef50.l | ⊢ ≤ = (le‘𝐾) |
cdlemef50.j | ⊢ ∨ = (join‘𝐾) |
cdlemef50.m | ⊢ ∧ = (meet‘𝐾) |
cdlemef50.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemef50.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemef50.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdlemef50.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdlemefs50.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemef50.f | ⊢ 𝐹 = (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 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) |
Ref | Expression |
---|---|
cdleme50lebi | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemef50.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemef50.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemef50.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemef50.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemef50.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemef50.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemef50.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | cdlemef50.d | . 2 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdlemefs50.e | . 2 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemef50.f | . 2 ⊢ 𝐹 = (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 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
11 | eqid 2734 | . 2 ⊢ ((𝑄 ∨ 𝑃) ∧ 𝑊) = ((𝑄 ∨ 𝑃) ∧ 𝑊) | |
12 | eqid 2734 | . 2 ⊢ ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) = ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) | |
13 | eqid 2734 | . 2 ⊢ ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))) = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))) | |
14 | eqid 2734 | . 2 ⊢ (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))))), ⦋𝑢 / 𝑣⦌((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))) ∨ (𝑎 ∧ 𝑊)))), 𝑎)) = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = ((𝑄 ∨ 𝑃) ∧ (((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊))))), ⦋𝑢 / 𝑣⦌((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))) ∨ (𝑎 ∧ 𝑊)))), 𝑎)) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | cdlemeg49lebilem 40521 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ⦋csb 3907 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 Basecbs 17244 lecple 17304 joincjn 18368 meetcmee 18369 Atomscatm 39244 HLchlt 39331 LHypclh 39966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-riotaBAD 38934 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-undef 8296 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-llines 39480 df-lplanes 39481 df-lvols 39482 df-lines 39483 df-psubsp 39485 df-pmap 39486 df-padd 39778 df-lhyp 39970 |
This theorem is referenced by: cdleme50eq 40523 cdleme50laut 40529 |
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