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Theorem dalem60 35800
Description: Lemma for dath 35804. 𝐵 is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
Hypotheses
Ref Expression
dalem.ph (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
dalem.l = (le‘𝐾)
dalem.j = (join‘𝐾)
dalem.a 𝐴 = (Atoms‘𝐾)
dalem.ps (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
dalem60.m = (meet‘𝐾)
dalem60.o 𝑂 = (LPlanes‘𝐾)
dalem60.y 𝑌 = ((𝑃 𝑄) 𝑅)
dalem60.z 𝑍 = ((𝑆 𝑇) 𝑈)
dalem60.d 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
dalem60.e 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
dalem60.g 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
dalem60.h 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
dalem60.i 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
dalem60.b1 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
Assertion
Ref Expression
dalem60 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)

Proof of Theorem dalem60
StepHypRef Expression
1 dalem.ph . . . 4 (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 𝐶 (𝑃 𝑄) ∧ ¬ 𝐶 (𝑄 𝑅) ∧ ¬ 𝐶 (𝑅 𝑃)) ∧ (¬ 𝐶 (𝑆 𝑇) ∧ ¬ 𝐶 (𝑇 𝑈) ∧ ¬ 𝐶 (𝑈 𝑆)) ∧ (𝐶 (𝑃 𝑆) ∧ 𝐶 (𝑄 𝑇) ∧ 𝐶 (𝑅 𝑈)))))
2 dalem.l . . . 4 = (le‘𝐾)
3 dalem.j . . . 4 = (join‘𝐾)
4 dalem.a . . . 4 𝐴 = (Atoms‘𝐾)
5 dalem.ps . . . 4 (𝜓 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (𝑐 𝑑))))
6 dalem60.m . . . 4 = (meet‘𝐾)
7 dalem60.o . . . 4 𝑂 = (LPlanes‘𝐾)
8 dalem60.y . . . 4 𝑌 = ((𝑃 𝑄) 𝑅)
9 dalem60.z . . . 4 𝑍 = ((𝑆 𝑇) 𝑈)
10 dalem60.d . . . 4 𝐷 = ((𝑃 𝑄) (𝑆 𝑇))
11 dalem60.g . . . 4 𝐺 = ((𝑐 𝑃) (𝑑 𝑆))
12 dalem60.h . . . 4 𝐻 = ((𝑐 𝑄) (𝑑 𝑇))
13 dalem60.i . . . 4 𝐼 = ((𝑐 𝑅) (𝑑 𝑈))
14 dalem60.b1 . . . 4 𝐵 = (((𝐺 𝐻) 𝐼) 𝑌)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14dalem57 35797 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)
16 dalem60.e . . . 4 𝐸 = ((𝑄 𝑅) (𝑇 𝑈))
171, 2, 3, 4, 5, 6, 7, 8, 9, 16, 11, 12, 13, 14dalem58 35798 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)
181dalemkelat 35692 . . . . 5 (𝜑𝐾 ∈ Lat)
19183ad2ant1 1167 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ Lat)
201, 2, 3, 4, 6, 7, 8, 9, 10dalemdea 35730 . . . . . 6 (𝜑𝐷𝐴)
21 eqid 2825 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2221, 4atbase 35357 . . . . . 6 (𝐷𝐴𝐷 ∈ (Base‘𝐾))
2320, 22syl 17 . . . . 5 (𝜑𝐷 ∈ (Base‘𝐾))
24233ad2ant1 1167 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐷 ∈ (Base‘𝐾))
251, 2, 3, 4, 6, 7, 8, 9, 16dalemeea 35731 . . . . . 6 (𝜑𝐸𝐴)
2621, 4atbase 35357 . . . . . 6 (𝐸𝐴𝐸 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 (𝜑𝐸 ∈ (Base‘𝐾))
28273ad2ant1 1167 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐸 ∈ (Base‘𝐾))
29 eqid 2825 . . . . . 6 (LLines‘𝐾) = (LLines‘𝐾)
301, 2, 3, 4, 5, 6, 29, 7, 8, 9, 11, 12, 13, 14dalem53 35793 . . . . 5 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (LLines‘𝐾))
3121, 29llnbase 35577 . . . . 5 (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
3230, 31syl 17 . . . 4 ((𝜑𝑌 = 𝑍𝜓) → 𝐵 ∈ (Base‘𝐾))
3321, 2, 3latjle12 17415 . . . 4 ((𝐾 ∈ Lat ∧ (𝐷 ∈ (Base‘𝐾) ∧ 𝐸 ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3419, 24, 28, 32, 33syl13anc 1495 . . 3 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐵𝐸 𝐵) ↔ (𝐷 𝐸) 𝐵))
3515, 17, 34mpbi2and 703 . 2 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) 𝐵)
361dalemkehl 35691 . . . 4 (𝜑𝐾 ∈ HL)
37363ad2ant1 1167 . . 3 ((𝜑𝑌 = 𝑍𝜓) → 𝐾 ∈ HL)
381, 2, 3, 4, 6, 7, 8, 9, 10, 16dalemdnee 35734 . . . . 5 (𝜑𝐷𝐸)
393, 4, 29llni2 35580 . . . . 5 (((𝐾 ∈ HL ∧ 𝐷𝐴𝐸𝐴) ∧ 𝐷𝐸) → (𝐷 𝐸) ∈ (LLines‘𝐾))
4036, 20, 25, 38, 39syl31anc 1496 . . . 4 (𝜑 → (𝐷 𝐸) ∈ (LLines‘𝐾))
41403ad2ant1 1167 . . 3 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) ∈ (LLines‘𝐾))
422, 29llncmp 35590 . . 3 ((𝐾 ∈ HL ∧ (𝐷 𝐸) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4337, 41, 30, 42syl3anc 1494 . 2 ((𝜑𝑌 = 𝑍𝜓) → ((𝐷 𝐸) 𝐵 ↔ (𝐷 𝐸) = 𝐵))
4435, 43mpbid 224 1 ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  meetcmee 17298  Latclat 17398  Atomscatm 35331  HLchlt 35418  LLinesclln 35559  LPlanesclpl 35560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35244  df-ol 35246  df-oml 35247  df-covers 35334  df-ats 35335  df-atl 35366  df-cvlat 35390  df-hlat 35419  df-llines 35566  df-lplanes 35567  df-lvols 35568
This theorem is referenced by:  dalem61  35801
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