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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk19w | Structured version Visualization version GIF version |
Description: Use a fixed element to eliminate 𝑃 in cdlemk19u 38100. (Contributed by NM, 1-Aug-2013.) |
Ref | Expression |
---|---|
cdlemk6.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemk6.j | ⊢ ∨ = (join‘𝐾) |
cdlemk6.m | ⊢ ∧ = (meet‘𝐾) |
cdlemk6.o | ⊢ ⊥ = (oc‘𝐾) |
cdlemk6.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemk6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemk6.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemk6.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
cdlemk6.p | ⊢ 𝑃 = ( ⊥ ‘𝑊) |
cdlemk6.z | ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) |
cdlemk6.y | ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) |
cdlemk6.x | ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) |
cdlemk6.u | ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) |
Ref | Expression |
---|---|
cdlemk19w | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈‘𝐹) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpb 1145 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) | |
2 | simp2 1133 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) | |
3 | eqid 2821 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
4 | cdlemk6.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
5 | cdlemk6.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | cdlemk6.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | 3, 4, 5, 6 | lhpocnel 37148 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊)) |
8 | 7 | 3ad2ant1 1129 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊)) |
9 | cdlemk6.p | . . . . 5 ⊢ 𝑃 = ( ⊥ ‘𝑊) | |
10 | 9 | eleq1i 2903 | . . . 4 ⊢ (𝑃 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐴) |
11 | 9 | breq1i 5065 | . . . . 5 ⊢ (𝑃(le‘𝐾)𝑊 ↔ ( ⊥ ‘𝑊)(le‘𝐾)𝑊) |
12 | 11 | notbii 322 | . . . 4 ⊢ (¬ 𝑃(le‘𝐾)𝑊 ↔ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊) |
13 | 10, 12 | anbi12i 628 | . . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊) ↔ (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊)) |
14 | 8, 13 | sylibr 236 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊)) |
15 | cdlemk6.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
16 | cdlemk6.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
17 | cdlemk6.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
18 | cdlemk6.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
19 | cdlemk6.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
20 | cdlemk6.z | . . 3 ⊢ 𝑍 = ((𝑃 ∨ (𝑅‘𝑏)) ∧ ((𝑁‘𝑃) ∨ (𝑅‘(𝑏 ∘ ◡𝐹)))) | |
21 | cdlemk6.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
22 | cdlemk6.x | . . 3 ⊢ 𝑋 = (℩𝑧 ∈ 𝑇 ∀𝑏 ∈ 𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝐹) ∧ (𝑅‘𝑏) ≠ (𝑅‘𝑔)) → (𝑧‘𝑃) = 𝑌)) | |
23 | cdlemk6.u | . . 3 ⊢ 𝑈 = (𝑔 ∈ 𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋)) | |
24 | 15, 3, 16, 17, 5, 6, 18, 19, 20, 21, 22, 23 | cdlemk19u 38100 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊)) → (𝑈‘𝐹) = 𝑁) |
25 | 1, 2, 14, 24 | syl3anc 1367 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝑈‘𝐹) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ifcif 4466 class class class wbr 5058 ↦ cmpt 5138 I cid 5453 ◡ccnv 5548 ↾ cres 5551 ∘ ccom 5553 ‘cfv 6349 ℩crio 7107 (class class class)co 7150 Basecbs 16477 lecple 16566 occoc 16567 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-undef 7933 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-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 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: cdlemk56w 38103 |
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