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Mathbox for Norm Megill |
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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 36991. (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 1181 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅‘𝐹) = (𝑅‘𝑁))) | |
2 | simp2 1168 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝑅‘𝐹) = (𝑅‘𝑁)) → (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇)) | |
3 | eqid 2799 | . . . . 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 36039 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊)) |
8 | 7 | 3ad2ant1 1164 | . . 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 2869 | . . . 4 ⊢ (𝑃 ∈ 𝐴 ↔ ( ⊥ ‘𝑊) ∈ 𝐴) |
11 | 9 | breq1i 4850 | . . . . 5 ⊢ (𝑃(le‘𝐾)𝑊 ↔ ( ⊥ ‘𝑊)(le‘𝐾)𝑊) |
12 | 11 | notbii 312 | . . . 4 ⊢ (¬ 𝑃(le‘𝐾)𝑊 ↔ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊) |
13 | 10, 12 | anbi12i 621 | . . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃(le‘𝐾)𝑊) ↔ (( ⊥ ‘𝑊) ∈ 𝐴 ∧ ¬ ( ⊥ ‘𝑊)(le‘𝐾)𝑊)) |
14 | 8, 13 | sylibr 226 | . 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 36991 | . 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 1491 | 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ifcif 4277 class class class wbr 4843 ↦ cmpt 4922 I cid 5219 ◡ccnv 5311 ↾ cres 5314 ∘ ccom 5316 ‘cfv 6101 ℩crio 6838 (class class class)co 6878 Basecbs 16184 lecple 16274 occoc 16275 joincjn 17259 meetcmee 17260 Atomscatm 35284 HLchlt 35371 LHypclh 36005 LTrncltrn 36122 trLctrl 36179 |
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 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-riotaBAD 34974 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-undef 7637 df-map 8097 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-llines 35519 df-lplanes 35520 df-lvols 35521 df-lines 35522 df-psubsp 35524 df-pmap 35525 df-padd 35817 df-lhyp 36009 df-laut 36010 df-ldil 36125 df-ltrn 36126 df-trl 36180 |
This theorem is referenced by: cdlemk56w 36994 |
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