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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemg17bq | Structured version Visualization version GIF version |
Description: cdlemg17b 37792 with 𝑃 and 𝑄 swapped. Antecedent 𝐹 ∈ (𝑇‘𝑊) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 37792 also? (Contributed by NM, 13-May-2013.) |
Ref | Expression |
---|---|
cdlemg12.l | ⊢ ≤ = (le‘𝐾) |
cdlemg12.j | ⊢ ∨ = (join‘𝐾) |
cdlemg12.m | ⊢ ∧ = (meet‘𝐾) |
cdlemg12.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemg12.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemg12.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemg12b.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
cdlemg17bq | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑄) = 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdlemg12.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | cdlemg12.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | cdlemg12.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | cdlemg12.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemg12.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemg12.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
7 | cdlemg12b.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
8 | 1, 2, 3, 4, 5, 6, 7 | cdlemg17pq 37802 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟))))) |
9 | simp11 1199 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | simp12 1200 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
11 | simp13 1201 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
12 | simp22 1203 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → 𝐺 ∈ 𝑇) | |
13 | simp23 1204 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → 𝑄 ≠ 𝑃) | |
14 | simp3 1134 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) | |
15 | 1, 2, 3, 4, 5, 6, 7 | cdlemg17b 37792 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → (𝐺‘𝑄) = 𝑃) |
16 | 9, 10, 11, 12, 13, 14, 15 | syl321anc 1388 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ≠ 𝑃) ∧ ((𝐺‘𝑄) ≠ 𝑄 ∧ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑃) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑄 ∨ 𝑟) = (𝑃 ∨ 𝑟)))) → (𝐺‘𝑄) = 𝑃) |
17 | 8, 16 | syl 17 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘𝑄) = 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 lecple 16566 joincjn 17548 meetcmee 17549 Atomscatm 36393 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 trLctrl 37288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 |
This theorem is referenced by: cdlemg17 37807 cdlemg18d 37811 |
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