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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme50trn123 | Structured version Visualization version GIF version | ||
| Description: Part of proof that 𝐹 is a translation. Combine all cases. TODO: fix comment. (Contributed by NM, 10-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 | 
|---|---|
| cdleme50trn123 | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cdlemef50.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | cdlemef50.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 3 | cdlemef50.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
| 4 | cdlemef50.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
| 5 | cdlemef50.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 6 | cdlemef50.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 7 | cdlemef50.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
| 8 | cdlemef50.d | . . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
| 9 | cdlemefs50.e | . . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
| 10 | cdlemef50.f | . . . 4 ⊢ 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 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50trn3 40556 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| 12 | 11 | anass1rs 655 | . 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑃 = 𝑄) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| 13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme50trn12 40555 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑃 ≠ 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| 14 | 13 | anass1rs 655 | . 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| 15 | 12, 14 | pm2.61dane 3028 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → ((𝑅 ∨ (𝐹‘𝑅)) ∧ 𝑊) = 𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ⦋csb 3898 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 ℩crio 7388 (class class class)co 7432 Basecbs 17248 lecple 17305 joincjn 18358 meetcmee 18359 Atomscatm 39265 HLchlt 39352 LHypclh 39987 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-riotaBAD 38955 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-undef 8299 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-llines 39501 df-lplanes 39502 df-lvols 39503 df-lines 39504 df-psubsp 39506 df-pmap 39507 df-padd 39799 df-lhyp 39991 | 
| This theorem is referenced by: cdleme50ltrn 40560 | 
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