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Theorem cdlemg16zz 40189
Description: Eliminate 𝑃 β‰  𝑄 from cdlemg16z 40188. TODO: Use this only if needed. (Contributed by NM, 26-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l ≀ = (leβ€˜πΎ)
cdlemg12.j ∨ = (joinβ€˜πΎ)
cdlemg12.m ∧ = (meetβ€˜πΎ)
cdlemg12.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg12.h 𝐻 = (LHypβ€˜πΎ)
cdlemg12.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemg12b.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemg16zz (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))

Proof of Theorem cdlemg16zz
StepHypRef Expression
1 id 22 . . . . 5 (𝑃 = 𝑄 β†’ 𝑃 = 𝑄)
2 2fveq3 6897 . . . . 5 (𝑃 = 𝑄 β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = (πΉβ€˜(πΊβ€˜π‘„)))
31, 2oveq12d 7434 . . . 4 (𝑃 = 𝑄 β†’ (𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) = (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))))
43oveq1d 7431 . . 3 (𝑃 = 𝑄 β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
54adantl 480 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 = 𝑄) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
6 simpl1 1188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
7 simpl21 1248 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
8 simpl22 1249 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
9 simpl23 1250 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ 𝐹 ∈ 𝑇)
10 simpl31 1251 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ 𝐺 ∈ 𝑇)
11 simpr 483 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 β‰  𝑄)
12 simpl32 1252 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄))
13 simpl33 1253 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))
14 cdlemg12.l . . . 4 ≀ = (leβ€˜πΎ)
15 cdlemg12.j . . . 4 ∨ = (joinβ€˜πΎ)
16 cdlemg12.m . . . 4 ∧ = (meetβ€˜πΎ)
17 cdlemg12.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
18 cdlemg12.h . . . 4 𝐻 = (LHypβ€˜πΎ)
19 cdlemg12.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
20 cdlemg12b.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
2114, 15, 16, 17, 18, 19, 20cdlemg16z 40188 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
226, 7, 8, 9, 10, 11, 12, 13, 21syl332anc 1398 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
235, 22pm2.61dane 3019 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  lecple 17239  joincjn 18302  meetcmee 18303  Atomscatm 38791  HLchlt 38878  LHypclh 39513  LTrncltrn 39630  trLctrl 39687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-riotaBAD 38481
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7991  df-2nd 7992  df-undef 8277  df-map 8845  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-clat 18490  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-llines 39027  df-lplanes 39028  df-lvols 39029  df-lines 39030  df-psubsp 39032  df-pmap 39033  df-padd 39325  df-lhyp 39517  df-laut 39518  df-ldil 39633  df-ltrn 39634  df-trl 39688
This theorem is referenced by:  cdlemg25zz  40219
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