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Theorem cdleme20aN 36111
Description: Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114. 𝐷, 𝐹, 𝑌, 𝐺 represent s2, f(s), t2, f(t). (Contributed by NM, 14-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme19.l = (le‘𝐾)
cdleme19.j = (join‘𝐾)
cdleme19.m = (meet‘𝐾)
cdleme19.a 𝐴 = (Atoms‘𝐾)
cdleme19.h 𝐻 = (LHyp‘𝐾)
cdleme19.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme19.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme19.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme19.d 𝐷 = ((𝑅 𝑆) 𝑊)
cdleme19.y 𝑌 = ((𝑅 𝑇) 𝑊)
cdleme20.v 𝑉 = ((𝑆 𝑇) 𝑊)
Assertion
Ref Expression
cdleme20aN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))

Proof of Theorem cdleme20aN
StepHypRef Expression
1 cdleme20.v . . 3 𝑉 = ((𝑆 𝑇) 𝑊)
21oveq1i 6802 . 2 (𝑉 𝐷) = (((𝑆 𝑇) 𝑊) 𝐷)
3 simp1l 1238 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
4 simp1r 1239 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
5 simp22 1248 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑆𝐴)
6 simp23 1249 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑆 𝑊)
7 simp21 1247 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
8 simp33 1252 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
9 simp32 1251 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
10 cdleme19.l . . . . . 6 = (le‘𝐾)
11 cdleme19.j . . . . . 6 = (join‘𝐾)
12 cdleme19.m . . . . . 6 = (meet‘𝐾)
13 cdleme19.a . . . . . 6 𝐴 = (Atoms‘𝐾)
14 cdleme19.h . . . . . 6 𝐻 = (LHyp‘𝐾)
15 cdleme19.d . . . . . 6 𝐷 = ((𝑅 𝑆) 𝑊)
1610, 11, 12, 13, 14, 15cdlemeda 36100 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑅𝐴𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐷𝐴)
173, 4, 5, 6, 7, 8, 9, 16syl223anc 1501 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐷𝐴)
18 simp31 1250 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑇𝐴)
19 eqid 2770 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2019, 11, 13hlatjcl 35168 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
213, 5, 18, 20syl3anc 1475 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 𝑇) ∈ (Base‘𝐾))
2219, 14lhpbase 35799 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
234, 22syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝑊 ∈ (Base‘𝐾))
24 hllat 35165 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ Lat)
253, 24syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
2619, 11, 13hlatjcl 35168 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
273, 7, 5, 26syl3anc 1475 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 𝑆) ∈ (Base‘𝐾))
2819, 10, 12latmle2 17284 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 𝑆) 𝑊) 𝑊)
2925, 27, 23, 28syl3anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑆) 𝑊) 𝑊)
3015, 29syl5eqbr 4819 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → 𝐷 𝑊)
3119, 10, 11, 12, 13atmod4i1 35667 . . . 4 ((𝐾 ∈ HL ∧ (𝐷𝐴 ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝐷 𝑊) → (((𝑆 𝑇) 𝑊) 𝐷) = (((𝑆 𝑇) 𝐷) 𝑊))
323, 17, 21, 23, 30, 31syl131anc 1488 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑆 𝑇) 𝑊) 𝐷) = (((𝑆 𝑇) 𝐷) 𝑊))
3310, 11, 12, 13, 14, 15cdleme10 36056 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑅𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝑆 𝐷) = (𝑆 𝑅))
343, 4, 7, 5, 6, 33syl212anc 1485 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 𝐷) = (𝑆 𝑅))
3534oveq1d 6807 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑆 𝐷) 𝑇) = ((𝑆 𝑅) 𝑇))
3611, 13hlatj32 35173 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝐷𝐴𝑇𝐴)) → ((𝑆 𝐷) 𝑇) = ((𝑆 𝑇) 𝐷))
373, 5, 17, 18, 36syl13anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑆 𝐷) 𝑇) = ((𝑆 𝑇) 𝐷))
3835, 37eqtr3d 2806 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → ((𝑆 𝑅) 𝑇) = ((𝑆 𝑇) 𝐷))
3938oveq1d 6807 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑆 𝑅) 𝑇) 𝑊) = (((𝑆 𝑇) 𝐷) 𝑊))
4032, 39eqtr4d 2807 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (((𝑆 𝑇) 𝑊) 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))
412, 40syl5eq 2816 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄))) → (𝑉 𝐷) = (((𝑆 𝑅) 𝑇) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144   class class class wbr 4784  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  Latclat 17252  Atomscatm 35065  HLchlt 35152  LHypclh 35785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789
This theorem is referenced by:  cdleme20bN  36112
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