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Theorem cdlemg6 40006
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 1188 . . 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 1223 . . 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 1224 . . 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 1251 . . 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 1252 . . 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 484 . . 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 1253 . . 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 2726 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
13 eqid 2726 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
14 eqid 2726 . . . 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 40005 . . 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 1390 . 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 1188 . . 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 1223 . . 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 1224 . . 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 1251 . . 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 1252 . . 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 484 . . 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 1253 . . 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 40000 . . 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 1390 . 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 810 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  lecple 17210  joincjn 18273  Atomscatm 38645  HLchlt 38732  LHypclh 39367  LTrncltrn 39484  trLctrl 39541
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-riotaBAD 38335
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-undef 8256  df-map 8821  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-llines 38881  df-lplanes 38882  df-lvols 38883  df-lines 38884  df-psubsp 38886  df-pmap 38887  df-padd 39179  df-lhyp 39371  df-laut 39372  df-ldil 39487  df-ltrn 39488  df-trl 39542
This theorem is referenced by:  cdlemg7aN  40008  cdlemg8a  40010  cdlemg8c  40012  cdlemg11a  40020
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