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Theorem cdleme43cN 36301
 Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 last line: r v g(s) = r v v2. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme43.b 𝐵 = (Base‘𝐾)
cdleme43.l = (le‘𝐾)
cdleme43.j = (join‘𝐾)
cdleme43.m = (meet‘𝐾)
cdleme43.a 𝐴 = (Atoms‘𝐾)
cdleme43.h 𝐻 = (LHyp‘𝐾)
cdleme43.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme43.x 𝑋 = ((𝑄 𝑃) 𝑊)
cdleme43.c 𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme43.f 𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))
cdleme43.d 𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))
cdleme43.g 𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))
cdleme43.e 𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))
cdleme43.v 𝑉 = ((𝑍 𝑆) 𝑊)
cdleme43.y 𝑌 = ((𝑅 𝐷) 𝑊)
Assertion
Ref Expression
cdleme43cN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑅 𝐷) = (𝑅 𝑌))

Proof of Theorem cdleme43cN
StepHypRef Expression
1 simp11 1245 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp22 1249 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
3 simp1 1130 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
4 simp21 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑄)
5 simp23 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
6 simp3 1132 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ 𝑆 (𝑃 𝑄))
7 cdleme43.b . . . 4 𝐵 = (Base‘𝐾)
8 cdleme43.l . . . 4 = (le‘𝐾)
9 cdleme43.j . . . 4 = (join‘𝐾)
10 cdleme43.m . . . 4 = (meet‘𝐾)
11 cdleme43.a . . . 4 𝐴 = (Atoms‘𝐾)
12 cdleme43.h . . . 4 𝐻 = (LHyp‘𝐾)
13 cdleme43.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
14 cdleme43.x . . . 4 𝑋 = ((𝑄 𝑃) 𝑊)
15 cdleme43.c . . . 4 𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
16 cdleme43.f . . . 4 𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))
17 cdleme43.d . . . 4 𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))
18 cdleme43.g . . . 4 𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))
19 cdleme43.e . . . 4 𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))
20 cdleme43.v . . . 4 𝑉 = ((𝑍 𝑆) 𝑊)
21 cdleme43.y . . . 4 𝑌 = ((𝑅 𝐷) 𝑊)
227, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21cdleme43bN 36300 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐷𝐴 ∧ ¬ 𝐷 𝑊))
233, 4, 5, 6, 22syl121anc 1481 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐷𝐴 ∧ ¬ 𝐷 𝑊))
247, 8, 9, 10, 11, 12, 21cdleme42a 36281 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝐷𝐴 ∧ ¬ 𝐷 𝑊)) → (𝑅 𝐷) = (𝑅 𝑌))
251, 2, 23, 24syl3anc 1476 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑅 𝐷) = (𝑅 𝑌))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4787  ‘cfv 6030  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35072  HLchlt 35159  LHypclh 35793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-lines 35310  df-psubsp 35312  df-pmap 35313  df-padd 35605  df-lhyp 35797 This theorem is referenced by:  cdlemeg46rjgN  36332
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