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Theorem cdleme21k 36119
Description: Eliminate 𝑆𝑇 condition in cdleme21 36118. (Contributed by NM, 26-Dec-2012.)
Hypotheses
Ref Expression
cdleme21.l = (le‘𝐾)
cdleme21.j = (join‘𝐾)
cdleme21.m = (meet‘𝐾)
cdleme21.a 𝐴 = (Atoms‘𝐾)
cdleme21.h 𝐻 = (LHyp‘𝐾)
cdleme21.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme21.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme21g.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme21g.d 𝐷 = ((𝑅 𝑆) 𝑊)
cdleme21g.y 𝑌 = ((𝑅 𝑇) 𝑊)
cdleme21g.n 𝑁 = ((𝑃 𝑄) (𝐹 𝐷))
cdleme21g.o 𝑂 = ((𝑃 𝑄) (𝐺 𝑌))
Assertion
Ref Expression
cdleme21k ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑁 = 𝑂)

Proof of Theorem cdleme21k
StepHypRef Expression
1 oveq1 6881 . . . . . . . 8 (𝑆 = 𝑇 → (𝑆 𝑈) = (𝑇 𝑈))
2 oveq2 6882 . . . . . . . . . 10 (𝑆 = 𝑇 → (𝑃 𝑆) = (𝑃 𝑇))
32oveq1d 6889 . . . . . . . . 9 (𝑆 = 𝑇 → ((𝑃 𝑆) 𝑊) = ((𝑃 𝑇) 𝑊))
43oveq2d 6890 . . . . . . . 8 (𝑆 = 𝑇 → (𝑄 ((𝑃 𝑆) 𝑊)) = (𝑄 ((𝑃 𝑇) 𝑊)))
51, 4oveq12d 6892 . . . . . . 7 (𝑆 = 𝑇 → ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊))))
6 cdleme21.f . . . . . . 7 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
7 cdleme21g.g . . . . . . 7 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
85, 6, 73eqtr4g 2865 . . . . . 6 (𝑆 = 𝑇𝐹 = 𝐺)
9 oveq2 6882 . . . . . . . 8 (𝑆 = 𝑇 → (𝑅 𝑆) = (𝑅 𝑇))
109oveq1d 6889 . . . . . . 7 (𝑆 = 𝑇 → ((𝑅 𝑆) 𝑊) = ((𝑅 𝑇) 𝑊))
11 cdleme21g.d . . . . . . 7 𝐷 = ((𝑅 𝑆) 𝑊)
12 cdleme21g.y . . . . . . 7 𝑌 = ((𝑅 𝑇) 𝑊)
1310, 11, 123eqtr4g 2865 . . . . . 6 (𝑆 = 𝑇𝐷 = 𝑌)
148, 13oveq12d 6892 . . . . 5 (𝑆 = 𝑇 → (𝐹 𝐷) = (𝐺 𝑌))
1514oveq2d 6890 . . . 4 (𝑆 = 𝑇 → ((𝑃 𝑄) (𝐹 𝐷)) = ((𝑃 𝑄) (𝐺 𝑌)))
16 cdleme21g.n . . . 4 𝑁 = ((𝑃 𝑄) (𝐹 𝐷))
17 cdleme21g.o . . . 4 𝑂 = ((𝑃 𝑄) (𝐺 𝑌))
1815, 16, 173eqtr4g 2865 . . 3 (𝑆 = 𝑇𝑁 = 𝑂)
1918eqeq1d 2808 . 2 (𝑆 = 𝑇 → (𝑁 = 𝑂𝑂 = 𝑂))
20 simpl11 1322 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
21 simpl12 1324 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
22 simpl13 1326 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
23 simpl21 1328 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
24 simpl22 1330 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
25 simpl23 1332 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
26 simpl3l 1294 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → 𝑃𝑄)
27 simpr 473 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → 𝑆𝑇)
2826, 27jca 503 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (𝑃𝑄𝑆𝑇))
29 simpl3r 1296 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))
30 cdleme21.l . . . 4 = (le‘𝐾)
31 cdleme21.j . . . 4 = (join‘𝐾)
32 cdleme21.m . . . 4 = (meet‘𝐾)
33 cdleme21.a . . . 4 𝐴 = (Atoms‘𝐾)
34 cdleme21.h . . . 4 𝐻 = (LHyp‘𝐾)
35 cdleme21.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
3630, 31, 32, 33, 34, 35, 6, 7, 11, 12, 16, 17cdleme21 36118 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ ((𝑃𝑄𝑆𝑇) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑁 = 𝑂)
3720, 21, 22, 23, 24, 25, 28, 29, 36syl332anc 1513 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) ∧ 𝑆𝑇) → 𝑁 = 𝑂)
38 eqidd 2807 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑂 = 𝑂)
3919, 37, 38pm2.61ne 3063 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄) ∧ 𝑅 (𝑃 𝑄)))) → 𝑁 = 𝑂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978   class class class wbr 4844  cfv 6101  (class class class)co 6874  lecple 16160  joincjn 17149  meetcmee 17150  Atomscatm 35043  HLchlt 35130  LHypclh 35764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-proset 17133  df-poset 17151  df-plt 17163  df-lub 17179  df-glb 17180  df-join 17181  df-meet 17182  df-p0 17244  df-p1 17245  df-lat 17251  df-clat 17313  df-oposet 34956  df-ol 34958  df-oml 34959  df-covers 35046  df-ats 35047  df-atl 35078  df-cvlat 35102  df-hlat 35131  df-llines 35278  df-lplanes 35279  df-lvols 35280  df-lines 35281  df-psubsp 35283  df-pmap 35284  df-padd 35576  df-lhyp 35768
This theorem is referenced by:  cdleme24  36133  cdleme43fsv1snlem  36201  cdleme37m  36243
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