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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg1a | Structured version Visualization version GIF version |
Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 36573 or vice-versa? Also, if not shortened with cdleme 36573, then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg1.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemg1.l | ⊢ ≤ = (le‘𝐾) |
cdlemg1.j | ⊢ ∨ = (join‘𝐾) |
cdlemg1.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg1.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
cdlemg1.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) |
cdlemg1.e | ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemg1.g | ⊢ 𝐺 = (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)) |
cdlemg1.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemg1a | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemg1.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg1.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg1.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg1.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg1.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg1.u | . . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | cdlemg1.d | . . . 4 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) | |
9 | cdlemg1.e | . . . 4 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemg1.g | . . . 4 ⊢ 𝐺 = (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 | cdlemg1.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | cdleme50ltrn 36570 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
13 | simpll1 1270 | . . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
14 | simplr 786 | . . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 ∈ 𝑇) | |
15 | 12 | ad2antrr 718 | . . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝐺 ∈ 𝑇) |
16 | simpll2 1272 | . . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
17 | simpr 478 | . . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑓‘𝑃) = 𝑄) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | cdleme17d 36511 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐺‘𝑃) = 𝑄) |
19 | 18 | ad2antrr 718 | . . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝐺‘𝑃) = 𝑄) |
20 | 17, 19 | eqtr4d 2834 | . . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → (𝑓‘𝑃) = (𝐺‘𝑃)) |
21 | 2, 5, 6, 11 | cdlemd 36220 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑓‘𝑃) = (𝐺‘𝑃)) → 𝑓 = 𝐺) |
22 | 13, 14, 15, 16, 20, 21 | syl311anc 1504 | . . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) ∧ (𝑓‘𝑃) = 𝑄) → 𝑓 = 𝐺) |
23 | 22 | ex 402 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 → 𝑓 = 𝐺)) |
24 | 18 | adantr 473 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝐺‘𝑃) = 𝑄) |
25 | fveq1 6408 | . . . . . 6 ⊢ (𝑓 = 𝐺 → (𝑓‘𝑃) = (𝐺‘𝑃)) | |
26 | 25 | eqeq1d 2799 | . . . . 5 ⊢ (𝑓 = 𝐺 → ((𝑓‘𝑃) = 𝑄 ↔ (𝐺‘𝑃) = 𝑄)) |
27 | 24, 26 | syl5ibrcom 239 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → (𝑓 = 𝐺 → (𝑓‘𝑃) = 𝑄)) |
28 | 23, 27 | impbid 204 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑓‘𝑃) = 𝑄 ↔ 𝑓 = 𝐺)) |
29 | 12, 28 | riota5 6863 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄) = 𝐺) |
30 | 29 | eqcomd 2803 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∀wral 3087 ⦋csb 3726 ifcif 4275 class class class wbr 4841 ↦ cmpt 4920 ‘cfv 6099 ℩crio 6836 (class class class)co 6876 Basecbs 16181 lecple 16271 joincjn 17256 meetcmee 17257 Atomscatm 35276 HLchlt 35363 LHypclh 35997 LTrncltrn 36114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-riotaBAD 34966 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-1st 7399 df-2nd 7400 df-undef 7635 df-map 8095 df-proset 17240 df-poset 17258 df-plt 17270 df-lub 17286 df-glb 17287 df-join 17288 df-meet 17289 df-p0 17351 df-p1 17352 df-lat 17358 df-clat 17420 df-oposet 35189 df-ol 35191 df-oml 35192 df-covers 35279 df-ats 35280 df-atl 35311 df-cvlat 35335 df-hlat 35364 df-llines 35511 df-lplanes 35512 df-lvols 35513 df-lines 35514 df-psubsp 35516 df-pmap 35517 df-padd 35809 df-lhyp 36001 df-laut 36002 df-ldil 36117 df-ltrn 36118 df-trl 36172 |
This theorem is referenced by: cdlemg1b2 36584 |
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