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Theorem cdlemd7 37207
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
Hypotheses
Ref Expression
cdlemd4.l = (le‘𝐾)
cdlemd4.j = (join‘𝐾)
cdlemd4.a 𝐴 = (Atoms‘𝐾)
cdlemd4.h 𝐻 = (LHyp‘𝐾)
cdlemd4.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemd7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑅) = (𝐺𝑅))

Proof of Theorem cdlemd7
StepHypRef Expression
1 simp1 1130 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴))
2 simp2l 1193 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp2r 1194 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp11l 1278 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝐾 ∈ HL)
54hllatd 36367 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝐾 ∈ Lat)
6 simp2rl 1236 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝑄𝐴)
7 eqid 2825 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
8 cdlemd4.a . . . . 5 𝐴 = (Atoms‘𝐾)
97, 8atbase 36292 . . . 4 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
106, 9syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝑄 ∈ (Base‘𝐾))
11 simp2ll 1234 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝑃𝐴)
127, 8atbase 36292 . . . 4 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1311, 12syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝑃 ∈ (Base‘𝐾))
14 simp11 1197 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp12l 1280 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝐹𝑇)
16 cdlemd4.h . . . . 5 𝐻 = (LHyp‘𝐾)
17 cdlemd4.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
187, 16, 17ltrncl 37128 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃 ∈ (Base‘𝐾)) → (𝐹𝑃) ∈ (Base‘𝐾))
1914, 15, 13, 18syl3anc 1365 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑃) ∈ (Base‘𝐾))
20 simp3r 1196 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → ¬ 𝑄 (𝑃 (𝐹𝑃)))
21 cdlemd4.l . . . . 5 = (le‘𝐾)
22 cdlemd4.j . . . . 5 = (join‘𝐾)
237, 21, 22latnlej1l 17671 . . . 4 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → 𝑄𝑃)
2423necomd 3075 . . 3 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝐹𝑃) ∈ (Base‘𝐾)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → 𝑃𝑄)
255, 10, 13, 19, 20, 24syl131anc 1377 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → 𝑃𝑄)
26 simp3l 1195 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑃) = (𝐺𝑃))
27 simp12 1198 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑇𝐺𝑇))
2821, 22, 8, 16, 17cdlemd6 37206 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑄) = (𝐺𝑄))
2914, 27, 2, 3, 20, 26, 28syl231anc 1384 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑄) = (𝐺𝑄))
3021, 22, 8, 16, 17cdlemd5 37205 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))
311, 2, 3, 25, 26, 29, 30syl132anc 1382 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑅) = (𝐺𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36266  HLchlt 36353  LHypclh 36987  LTrncltrn 37104
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-map 8401  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991  df-laut 36992  df-ldil 37107  df-ltrn 37108  df-trl 37162
This theorem is referenced by:  cdlemd9  37209
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