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Theorem cdlemd6 35993
 Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-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
cdlemd6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑄) = (𝐺𝑄))

Proof of Theorem cdlemd6
StepHypRef Expression
1 simp3 1133 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑃) = (𝐺𝑃))
21oveq2d 6829 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝑃 (𝐹𝑃)) = (𝑃 (𝐺𝑃)))
32oveq1d 6828 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊) = ((𝑃 (𝐺𝑃))(meet‘𝐾)𝑊))
4 simp1l 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simp1rl 1305 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → 𝐹𝑇)
6 simp21 1249 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
7 cdlemd4.l . . . . . . 7 = (le‘𝐾)
8 cdlemd4.j . . . . . . 7 = (join‘𝐾)
9 eqid 2760 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
10 cdlemd4.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
11 cdlemd4.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
12 cdlemd4.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
13 eqid 2760 . . . . . . 7 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
147, 8, 9, 10, 11, 12, 13trlval2 35953 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((trL‘𝐾)‘𝑊)‘𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
154, 5, 6, 14syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐹) = ((𝑃 (𝐹𝑃))(meet‘𝐾)𝑊))
16 simp1rr 1306 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → 𝐺𝑇)
177, 8, 9, 10, 11, 12, 13trlval2 35953 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (((trL‘𝐾)‘𝑊)‘𝐺) = ((𝑃 (𝐺𝑃))(meet‘𝐾)𝑊))
184, 16, 6, 17syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐺) = ((𝑃 (𝐺𝑃))(meet‘𝐾)𝑊))
193, 15, 183eqtr4d 2804 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (((trL‘𝐾)‘𝑊)‘𝐹) = (((trL‘𝐾)‘𝑊)‘𝐺))
2019oveq2d 6829 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝑄 (((trL‘𝐾)‘𝑊)‘𝐹)) = (𝑄 (((trL‘𝐾)‘𝑊)‘𝐺)))
211oveq1d 6828 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → ((𝐹𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊)) = ((𝐺𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊)))
2220, 21oveq12d 6831 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))) = ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))))
23 simp22 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
24 simp23 1251 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → ¬ 𝑄 (𝑃 (𝐹𝑃)))
257, 8, 9, 10, 11, 12, 13cdlemc 35987 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝐹𝑄) = ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))))
264, 5, 6, 23, 24, 25syl131anc 1490 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑄) = ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐹))(meet‘𝐾)((𝐹𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))))
27 oveq2 6821 . . . . . . 7 ((𝐹𝑃) = (𝐺𝑃) → (𝑃 (𝐹𝑃)) = (𝑃 (𝐺𝑃)))
2827breq2d 4816 . . . . . 6 ((𝐹𝑃) = (𝐺𝑃) → (𝑄 (𝑃 (𝐹𝑃)) ↔ 𝑄 (𝑃 (𝐺𝑃))))
2928notbid 307 . . . . 5 ((𝐹𝑃) = (𝐺𝑃) → (¬ 𝑄 (𝑃 (𝐹𝑃)) ↔ ¬ 𝑄 (𝑃 (𝐺𝑃))))
3029biimpd 219 . . . 4 ((𝐹𝑃) = (𝐺𝑃) → (¬ 𝑄 (𝑃 (𝐹𝑃)) → ¬ 𝑄 (𝑃 (𝐺𝑃))))
311, 24, 30sylc 65 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → ¬ 𝑄 (𝑃 (𝐺𝑃)))
327, 8, 9, 10, 11, 12, 13cdlemc 35987 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐺𝑃))) → (𝐺𝑄) = ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))))
334, 16, 6, 23, 31, 32syl131anc 1490 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐺𝑄) = ((𝑄 (((trL‘𝐾)‘𝑊)‘𝐺))(meet‘𝐾)((𝐺𝑃) ((𝑃 𝑄)(meet‘𝐾)𝑊))))
3422, 26, 333eqtr4d 2804 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑄) = (𝐺𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   class class class wbr 4804  ‘cfv 6049  (class class class)co 6813  lecple 16150  joincjn 17145  meetcmee 17146  Atomscatm 35053  HLchlt 35140  LHypclh 35773  LTrncltrn 35890  trLctrl 35948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-psubsp 35292  df-pmap 35293  df-padd 35585  df-lhyp 35777  df-laut 35778  df-ldil 35893  df-ltrn 35894  df-trl 35949 This theorem is referenced by:  cdlemd7  35994
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