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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg2k | Structured version Visualization version GIF version |
Description: cdleme42keg 37624 with simpler hypotheses. TODO: FIX COMMENT. TODO: derive from cdlemg3a 37735, cdlemg2fv2 37738, cdlemg2jOLDN 37736, ltrnel 37277? (Contributed by NM, 22-Apr-2013.) |
Ref | Expression |
---|---|
cdlemg2inv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg2inv.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg2j.l | ⊢ ≤ = (le‘𝐾) |
cdlemg2j.j | ⊢ ∨ = (join‘𝐾) |
cdlemg2j.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg2j.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg2j.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
Ref | Expression |
---|---|
cdlemg2k | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . 2 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | cdlemg2j.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | cdlemg2j.j | . 2 ⊢ ∨ = (join‘𝐾) | |
4 | cdlemg2j.m | . 2 ⊢ ∧ = (meet‘𝐾) | |
5 | cdlemg2j.a | . 2 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemg2inv.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | cdlemg2inv.t | . 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | eqid 2823 | . 2 ⊢ ((𝑝 ∨ 𝑞) ∧ 𝑊) = ((𝑝 ∨ 𝑞) ∧ 𝑊) | |
9 | eqid 2823 | . 2 ⊢ ((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) = ((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) | |
10 | eqid 2823 | . 2 ⊢ ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
11 | eqid 2823 | . 2 ⊢ (𝑥 ∈ (Base‘𝐾) ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) = (𝑥 ∈ (Base‘𝐾) ↦ if((𝑝 ≠ 𝑞 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ (Base‘𝐾)∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑝 ∨ 𝑞), (℩𝑦 ∈ (Base‘𝐾)∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑝 ∨ 𝑞)) → 𝑦 = ((𝑝 ∨ 𝑞) ∧ (((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))))), ⦋𝑠 / 𝑡⦌((𝑡 ∨ ((𝑝 ∨ 𝑞) ∧ 𝑊)) ∧ (𝑞 ∨ ((𝑝 ∨ 𝑡) ∧ 𝑊)))) ∨ (𝑥 ∧ 𝑊)))), 𝑥)) | |
12 | cdlemg2j.u | . 2 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | cdlemg2klem 37733 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝐹 ∈ 𝑇) → ((𝐹‘𝑃) ∨ (𝐹‘𝑄)) = ((𝐹‘𝑃) ∨ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⦋csb 3885 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 meetcmee 17557 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-undef 7941 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 |
This theorem is referenced by: cdlemg2kq 37740 cdlemg2l 37741 cdlemg2m 37742 cdlemg9b 37771 cdlemg10bALTN 37774 cdlemg12b 37782 cdlemg17e 37803 |
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