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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2klem | Structured version Visualization version GIF version |
Description: cdleme42keg 36635 with simpler hypotheses. TODO: FIX COMMENT. (Contributed by NM, 22-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg2.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemg2.l | ⊢ ≤ = (le‘𝐾) |
cdlemg2.j | ⊢ ∨ = (join‘𝐾) |
cdlemg2.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg2ex.u | ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊) |
cdlemg2ex.d | ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) |
cdlemg2ex.e | ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdlemg2ex.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)) |
cdlemg2klem.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
Ref | Expression |
---|---|
cdlemg2klem | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cdlemg2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg2.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg2.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg2.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | cdlemg2ex.u | . . 3 ⊢ 𝑈 = ((𝑝 ∨ 𝑞) ∧ 𝑊) | |
9 | cdlemg2ex.d | . . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) | |
10 | cdlemg2ex.e | . . 3 ⊢ 𝐸 = ((𝑝 ∨ 𝑞) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
11 | cdlemg2ex.g | . . 3 ⊢ 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)) | |
12 | fveq1 6445 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑃) = (𝐺‘𝑃)) | |
13 | fveq1 6445 | . . . . 5 ⊢ (𝐹 = 𝐺 → (𝐹‘𝑄) = (𝐺‘𝑄)) | |
14 | 12, 13 | oveq12d 6940 | . . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐺‘𝑃) ∨ (𝐺‘𝑄))) |
15 | 12 | oveq1d 6937 | . . . 4 ⊢ (𝐹 = 𝐺 → ((𝐹‘𝑃) ∨ 𝑉) = ((𝐺‘𝑃) ∨ 𝑉)) |
16 | 14, 15 | eqeq12d 2792 | . . 3 ⊢ (𝐹 = 𝐺 → (((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉) ↔ ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉))) |
17 | vex 3400 | . . . . 5 ⊢ 𝑠 ∈ V | |
18 | eqid 2777 | . . . . . 6 ⊢ ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) | |
19 | 9, 18 | cdleme31sc 36533 | . . . . 5 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊)))) |
20 | 17, 19 | ax-mp 5 | . . . 4 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑠) ∧ 𝑊))) |
21 | eqid 2777 | . . . 4 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)) | |
22 | eqid 2777 | . . . 4 ⊢ if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) | |
23 | eqid 2777 | . . . 4 ⊢ (℩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)))) = (℩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)))) | |
24 | cdlemg2klem.v | . . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
25 | 1, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11, 24 | cdleme42keg 36635 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐺‘𝑃) ∨ (𝐺‘𝑄)) = ((𝐺‘𝑃) ∨ 𝑉)) |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25 | cdlemg2ce 36741 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
27 | 26 | 3com23 1117 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∀wral 3089 Vcvv 3397 ⦋csb 3750 ifcif 4306 class class class wbr 4886 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 Basecbs 16255 lecple 16345 joincjn 17330 meetcmee 17331 Atomscatm 35412 HLchlt 35499 LHypclh 36133 LTrncltrn 36250 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-riotaBAD 35102 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-undef 7681 df-map 8142 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-oposet 35325 df-ol 35327 df-oml 35328 df-covers 35415 df-ats 35416 df-atl 35447 df-cvlat 35471 df-hlat 35500 df-llines 35647 df-lplanes 35648 df-lvols 35649 df-lines 35650 df-psubsp 35652 df-pmap 35653 df-padd 35945 df-lhyp 36137 df-laut 36138 df-ldil 36253 df-ltrn 36254 df-trl 36308 |
This theorem is referenced by: cdlemg2k 36750 |
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