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Theorem cdlemg6e 40151
Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l ≀ = (leβ€˜πΎ)
cdlemg4.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg4.h 𝐻 = (LHypβ€˜πΎ)
cdlemg4.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemg4.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
cdlemg4.j ∨ = (joinβ€˜πΎ)
cdlemg4b.v 𝑉 = (π‘…β€˜πΊ)
Assertion
Ref Expression
cdlemg6e (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
Proof of Theorem cdlemg6e
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 simp1 1133 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simp21 1203 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
3 simp31 1206 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ 𝐺 ∈ 𝑇)
4 cdlemg4.l . . . . 5 ≀ = (leβ€˜πΎ)
5 cdlemg4.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
6 cdlemg4.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
7 cdlemg4.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
84, 5, 6, 7ltrnel 39668 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΊβ€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (πΊβ€˜π‘ƒ) ≀ π‘Š))
91, 3, 2, 8syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ ((πΊβ€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (πΊβ€˜π‘ƒ) ≀ π‘Š))
10 cdlemg4.j . . . 4 ∨ = (joinβ€˜πΎ)
114, 10, 5, 6cdlemb3 40135 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ ((πΊβ€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (πΊβ€˜π‘ƒ) ≀ π‘Š)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ))))
121, 2, 9, 11syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ))))
13 cdlemg4.r . . . . . 6 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
14 cdlemg4b.v . . . . . 6 𝑉 = (π‘…β€˜πΊ)
154, 5, 6, 7, 13, 10, 14cdlemg6d 40150 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (((π‘Ÿ ∈ 𝐴 ∧ Β¬ π‘Ÿ ≀ π‘Š) ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄))
1615exp4c 431 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (π‘Ÿ ∈ 𝐴 β†’ (Β¬ π‘Ÿ ≀ π‘Š β†’ (Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄))))
1716imp4a 421 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (π‘Ÿ ∈ 𝐴 β†’ ((Β¬ π‘Ÿ ≀ π‘Š ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)))
1817rexlimdv 3143 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ (πΊβ€˜π‘ƒ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄))
1912, 18mpd 15 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃 ∨ 𝑉) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   class class class wbr 5143  β€˜cfv 6543  (class class class)co 7416  lecple 17239  joincjn 18302  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:  cdlemg6  40152
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