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Theorem cdlemg4d 37867
Description: TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
Hypotheses
Ref Expression
cdlemg4.l = (le‘𝐾)
cdlemg4.a 𝐴 = (Atoms‘𝐾)
cdlemg4.h 𝐻 = (LHyp‘𝐾)
cdlemg4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg4.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemg4.j = (join‘𝐾)
cdlemg4b.v 𝑉 = (𝑅𝐺)
Assertion
Ref Expression
cdlemg4d (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ (𝐺𝑄) ((𝐺𝑃) (𝐹‘(𝐺𝑃))))

Proof of Theorem cdlemg4d
StepHypRef Expression
1 simp1 1133 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1203 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp22 1204 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp31 1206 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐺𝑇)
5 simp32 1207 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ 𝑄 (𝑃 𝑉))
6 cdlemg4.l . . . 4 = (le‘𝐾)
7 cdlemg4.a . . . 4 𝐴 = (Atoms‘𝐾)
8 cdlemg4.h . . . 4 𝐻 = (LHyp‘𝐾)
9 cdlemg4.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
10 cdlemg4.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
11 cdlemg4.j . . . 4 = (join‘𝐾)
12 cdlemg4b.v . . . 4 𝑉 = (𝑅𝐺)
136, 7, 8, 9, 10, 11, 12cdlemg4c 37866 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) ∧ ¬ 𝑄 (𝑃 𝑉)) → ¬ (𝐺𝑄) (𝑃 𝑉))
141, 2, 3, 4, 5, 13syl131anc 1380 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ (𝐺𝑄) (𝑃 𝑉))
15 simp1l 1194 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐾 ∈ HL)
16 simp21l 1287 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑃𝐴)
176, 7, 8, 9ltrnel 37393 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
1817simpld 498 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝑃) ∈ 𝐴)
191, 4, 2, 18syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐺𝑃) ∈ 𝐴)
2011, 7hlatjcom 36622 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐺𝑃) ∈ 𝐴) → (𝑃 (𝐺𝑃)) = ((𝐺𝑃) 𝑃))
2115, 16, 19, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑃 (𝐺𝑃)) = ((𝐺𝑃) 𝑃))
226, 7, 8, 9, 10, 11, 12cdlemg4b1 37863 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺𝑇) → (𝑃 𝑉) = (𝑃 (𝐺𝑃)))
231, 2, 4, 22syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑃 𝑉) = (𝑃 (𝐺𝑃)))
24 simp33 1208 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑃)) = 𝑃)
2524oveq2d 7156 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ((𝐺𝑃) (𝐹‘(𝐺𝑃))) = ((𝐺𝑃) 𝑃))
2621, 23, 253eqtr4rd 2868 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ((𝐺𝑃) (𝐹‘(𝐺𝑃))) = (𝑃 𝑉))
2726breq2d 5054 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ((𝐺𝑄) ((𝐺𝑃) (𝐹‘(𝐺𝑃))) ↔ (𝐺𝑄) (𝑃 𝑉)))
2814, 27mtbird 328 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ (𝐺𝑄) ((𝐺𝑃) (𝐹‘(𝐺𝑃))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114   class class class wbr 5042  cfv 6334  (class class class)co 7140  lecple 16563  joincjn 17545  Atomscatm 36517  HLchlt 36604  LHypclh 37238  LTrncltrn 37355  trLctrl 37412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36430  df-ol 36432  df-oml 36433  df-covers 36520  df-ats 36521  df-atl 36552  df-cvlat 36576  df-hlat 36605  df-psubsp 36757  df-pmap 36758  df-padd 37050  df-lhyp 37242  df-laut 37243  df-ldil 37358  df-ltrn 37359  df-trl 37413
This theorem is referenced by:  cdlemg4e  37868
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