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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 2737 | . 2 ⊢ ((𝑄 ∨ 𝑃) ∧ 𝑊) = ((𝑄 ∨ 𝑃) ∧ 𝑊) | |
| 12 | eqid 2737 | . 2 ⊢ ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) = ((𝑣 ∨ ((𝑄 ∨ 𝑃) ∧ 𝑊)) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊))) | |
| 13 | eqid 2737 | . 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 2737 | . 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 40541 | 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⦋csb 3899 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 HLchlt 39351 LHypclh 39986 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 |
| This theorem is referenced by: cdleme50eq 40543 cdleme50laut 40549 |
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