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Theorem cdlemg4d 38635
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 1135 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp21 1205 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp22 1206 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp31 1208 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐺𝑇)
5 simp32 1209 . . 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 38634 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) ∧ ¬ 𝑄 (𝑃 𝑉)) → ¬ (𝐺𝑄) (𝑃 𝑉))
141, 2, 3, 4, 5, 13syl131anc 1382 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ (𝐺𝑄) (𝑃 𝑉))
15 simp1l 1196 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝐾 ∈ HL)
16 simp21l 1289 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → 𝑃𝐴)
176, 7, 8, 9ltrnel 38161 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
1817simpld 495 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝑃) ∈ 𝐴)
191, 4, 2, 18syl3anc 1370 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐺𝑃) ∈ 𝐴)
2011, 7hlatjcom 37390 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴 ∧ (𝐺𝑃) ∈ 𝐴) → (𝑃 (𝐺𝑃)) = ((𝐺𝑃) 𝑃))
2115, 16, 19, 20syl3anc 1370 . . . 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 38631 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺𝑇) → (𝑃 𝑉) = (𝑃 (𝐺𝑃)))
231, 2, 4, 22syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝑃 𝑉) = (𝑃 (𝐺𝑃)))
24 simp33 1210 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑃)) = 𝑃)
2524oveq2d 7283 . . . 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 2789 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ((𝐺𝑃) (𝐹‘(𝐺𝑃))) = (𝑃 𝑉))
2726breq2d 5085 . 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 325 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 396  w3a 1086   = wceq 1539  wcel 2106   class class class wbr 5073  cfv 6426  (class class class)co 7267  lecple 16979  joincjn 18039  Atomscatm 37285  HLchlt 37372  LHypclh 38006  LTrncltrn 38123  trLctrl 38180
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5208  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-1st 7820  df-2nd 7821  df-map 8604  df-proset 18023  df-poset 18041  df-plt 18058  df-lub 18074  df-glb 18075  df-join 18076  df-meet 18077  df-p0 18153  df-p1 18154  df-lat 18160  df-clat 18227  df-oposet 37198  df-ol 37200  df-oml 37201  df-covers 37288  df-ats 37289  df-atl 37320  df-cvlat 37344  df-hlat 37373  df-psubsp 37525  df-pmap 37526  df-padd 37818  df-lhyp 38010  df-laut 38011  df-ldil 38126  df-ltrn 38127  df-trl 38181
This theorem is referenced by:  cdlemg4e  38636
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