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Theorem cdlemg6 39494
Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
Hypotheses
Ref Expression
cdlemg6.l ≀ = (leβ€˜πΎ)
cdlemg6.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg6.h 𝐻 = (LHypβ€˜πΎ)
cdlemg6.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
cdlemg6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
Proof of Theorem cdlemg6
StepHypRef Expression
1 simpl1 1192 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simpl2l 1227 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
3 simpl2r 1228 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
4 simpl31 1255 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐹 ∈ 𝑇)
5 simpl32 1256 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐺 ∈ 𝑇)
6 simpr 486 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)))
7 simpl33 1257 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)
8 cdlemg6.l . . . 4 ≀ = (leβ€˜πΎ)
9 cdlemg6.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
10 cdlemg6.h . . . 4 𝐻 = (LHypβ€˜πΎ)
11 cdlemg6.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
12 eqid 2733 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
13 eqid 2733 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2733 . . . 4 (((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ) = (((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)
158, 9, 10, 11, 12, 13, 14cdlemg6e 39493 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1394 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
17 simpl1 1192 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 simpl2l 1227 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
19 simpl2r 1228 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
20 simpl31 1255 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐹 ∈ 𝑇)
21 simpl32 1256 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ 𝐺 ∈ 𝑇)
22 simpr 486 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)))
23 simpl33 1257 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)
248, 9, 10, 11, 12, 13, 14cdlemg4 39488 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ)) ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
2517, 18, 19, 20, 21, 22, 23, 24syl133anc 1394 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜(πΊβ€˜π‘ƒ)) = 𝑃)) ∧ Β¬ 𝑄 ≀ (𝑃(joinβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = 𝑄)
2616, 25pm2.61dan 812 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  lecple 17204  joincjn 18264  Atomscatm 38133  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  trLctrl 39029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030
This theorem is referenced by:  cdlemg7aN  39496  cdlemg8a  39498  cdlemg8c  39500  cdlemg11a  39508
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