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Theorem cdleme38m 36253
 Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013.)
Hypotheses
Ref Expression
cdleme38.l = (le‘𝐾)
cdleme38.j = (join‘𝐾)
cdleme38.m = (meet‘𝐾)
cdleme38.a 𝐴 = (Atoms‘𝐾)
cdleme38.h 𝐻 = (LHyp‘𝐾)
cdleme38.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme38.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme38.d 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
cdleme38.v 𝑉 = ((𝑡 𝐸) 𝑊)
cdleme38.x 𝑋 = ((𝑢 𝐷) 𝑊)
cdleme38.f 𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))
cdleme38.g 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme38m ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 = 𝑆)

Proof of Theorem cdleme38m
StepHypRef Expression
1 simp1 1131 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
2 simp2 1132 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)))
3 simp311 1405 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 (𝑃 𝑄))
4 simp312 1406 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑆 (𝑃 𝑄))
5 simp313 1407 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐹 = 𝐺)
63, 4jca 555 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)))
7 simp32 1253 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)))
8 simp33 1254 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))
9 cdleme38.l . . . . . 6 = (le‘𝐾)
10 cdleme38.j . . . . . 6 = (join‘𝐾)
11 cdleme38.m . . . . . 6 = (meet‘𝐾)
12 cdleme38.a . . . . . 6 𝐴 = (Atoms‘𝐾)
13 cdleme38.h . . . . . 6 𝐻 = (LHyp‘𝐾)
14 cdleme38.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
15 cdleme38.e . . . . . 6 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
16 cdleme38.d . . . . . 6 𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
17 cdleme38.v . . . . . 6 𝑉 = ((𝑡 𝐸) 𝑊)
18 cdleme38.x . . . . . 6 𝑋 = ((𝑢 𝐷) 𝑊)
19 eqid 2760 . . . . . 6 ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊))) = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))
20 cdleme38.g . . . . . 6 𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))
219, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme37m 36252 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊))) = 𝐺)
221, 2, 6, 7, 8, 21syl113anc 1489 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊))) = 𝐺)
235, 22eqtr4d 2797 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐹 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊))))
243, 4, 233jca 1123 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))))
25 eqid 2760 . . 3 (Base‘𝐾) = (Base‘𝐾)
26 cdleme38.f . . 3 𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))
2725, 9, 10, 11, 12, 13, 14, 15, 17, 26, 19cdleme36m 36251 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)))) → 𝑅 = 𝑆)
281, 2, 24, 7, 27syl112anc 1481 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 = 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932   class class class wbr 4804  ‘cfv 6049  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  Atomscatm 35053  HLchlt 35140  LHypclh 35773 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-3or 1073  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-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-lplanes 35288  df-lvols 35289  df-lines 35290  df-psubsp 35292  df-pmap 35293  df-padd 35585  df-lhyp 35777 This theorem is referenced by:  cdleme38n  36254
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