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Theorem cdleme21k 37997
Description: Eliminate 𝑆𝑇 condition in cdleme21 37996. (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 7179 . . . . . . . 8 (𝑆 = 𝑇 → (𝑆 𝑈) = (𝑇 𝑈))
2 oveq2 7180 . . . . . . . . . 10 (𝑆 = 𝑇 → (𝑃 𝑆) = (𝑃 𝑇))
32oveq1d 7187 . . . . . . . . 9 (𝑆 = 𝑇 → ((𝑃 𝑆) 𝑊) = ((𝑃 𝑇) 𝑊))
43oveq2d 7188 . . . . . . . 8 (𝑆 = 𝑇 → (𝑄 ((𝑃 𝑆) 𝑊)) = (𝑄 ((𝑃 𝑇) 𝑊)))
51, 4oveq12d 7190 . . . . . . 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 2798 . . . . . 6 (𝑆 = 𝑇𝐹 = 𝐺)
9 oveq2 7180 . . . . . . . 8 (𝑆 = 𝑇 → (𝑅 𝑆) = (𝑅 𝑇))
109oveq1d 7187 . . . . . . 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 2798 . . . . . 6 (𝑆 = 𝑇𝐷 = 𝑌)
148, 13oveq12d 7190 . . . . 5 (𝑆 = 𝑇 → (𝐹 𝐷) = (𝐺 𝑌))
1514oveq2d 7188 . . . 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 2798 . . 3 (𝑆 = 𝑇𝑁 = 𝑂)
1918eqeq1d 2740 . 2 (𝑆 = 𝑇 → (𝑁 = 𝑂𝑂 = 𝑂))
20 simpl11 1249 . . 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 1250 . . 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 1251 . . 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 1252 . . 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 1253 . . 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 1254 . . 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 1229 . . . 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 488 . . . 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 515 . . 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 1230 . . 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 37996 . . 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 1402 . 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 2739 . 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 3019 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 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934   class class class wbr 5030  cfv 6339  (class class class)co 7172  lecple 16677  joincjn 17672  meetcmee 17673  Atomscatm 36922  HLchlt 37009  LHypclh 37643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-1st 7716  df-2nd 7717  df-proset 17656  df-poset 17674  df-plt 17686  df-lub 17702  df-glb 17703  df-join 17704  df-meet 17705  df-p0 17767  df-p1 17768  df-lat 17774  df-clat 17836  df-oposet 36835  df-ol 36837  df-oml 36838  df-covers 36925  df-ats 36926  df-atl 36957  df-cvlat 36981  df-hlat 37010  df-llines 37157  df-lplanes 37158  df-lvols 37159  df-lines 37160  df-psubsp 37162  df-pmap 37163  df-padd 37455  df-lhyp 37647
This theorem is referenced by:  cdleme24  38011  cdleme43fsv1snlem  38079  cdleme37m  38121
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