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Theorem arglem1N 37206
Description: Lemma for Desargues's law. Theorem 13.3 of [Crawley] p. 110, third and fourth lines from bottom. In these lemmas, 𝑃, 𝑄, 𝑅, 𝑆, 𝑇, 𝑈, 𝐶, 𝐷, 𝐸, 𝐹, and 𝐺 represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
arglem1.j = (join‘𝐾)
arglem1.m = (meet‘𝐾)
arglem1.a 𝐴 = (Atoms‘𝐾)
arglem1.f 𝐹 = ((𝑃 𝑄) (𝑆 𝑇))
arglem1.g 𝐺 = ((𝑃 𝑆) (𝑄 𝑇))
Assertion
Ref Expression
arglem1N ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐹𝐴)

Proof of Theorem arglem1N
StepHypRef Expression
1 arglem1.f . 2 𝐹 = ((𝑃 𝑄) (𝑆 𝑇))
2 simpl11 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ HL)
32hllatd 36380 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ Lat)
4 simpl12 1241 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝐴)
5 eqid 2818 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
6 arglem1.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6atbase 36305 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
84, 7syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃 ∈ (Base‘𝐾))
9 simpl13 1242 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝐴)
105, 6atbase 36305 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
119, 10syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄 ∈ (Base‘𝐾))
12 simpl21 1243 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝐴)
135, 6atbase 36305 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆 ∈ (Base‘𝐾))
15 simpl22 1244 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑇𝐴)
165, 6atbase 36305 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1715, 16syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑇 ∈ (Base‘𝐾))
18 arglem1.j . . . . . 6 = (join‘𝐾)
195, 18latj4 17699 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾))) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
203, 8, 11, 14, 17, 19syl122anc 1371 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
21 arglem1.g . . . . . 6 𝐺 = ((𝑃 𝑆) (𝑄 𝑇))
22 simpr 485 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐺𝐴)
2321, 22eqeltrrid 2915 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
24 simpl31 1246 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑆)
25 eqid 2818 . . . . . . . 8 (LLines‘𝐾) = (LLines‘𝐾)
2618, 6, 25llni2 36528 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) ∧ 𝑃𝑆) → (𝑃 𝑆) ∈ (LLines‘𝐾))
272, 4, 12, 24, 26syl31anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑆) ∈ (LLines‘𝐾))
28 simpl32 1247 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝑇)
2918, 6, 25llni2 36528 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
302, 9, 15, 28, 29syl31anc 1365 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑄 𝑇) ∈ (LLines‘𝐾))
31 arglem1.m . . . . . . 7 = (meet‘𝐾)
32 eqid 2818 . . . . . . 7 (LPlanes‘𝐾) = (LPlanes‘𝐾)
3318, 31, 6, 25, 322llnmj 36576 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾)))
342, 27, 30, 33syl3anc 1363 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾)))
3523, 34mpbid 233 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾))
3620, 35eqeltrd 2910 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾))
37 simpl23 1245 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑄)
3818, 6, 25llni2 36528 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
392, 4, 9, 37, 38syl31anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑄) ∈ (LLines‘𝐾))
40 simpl33 1248 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝑇)
4118, 6, 25llni2 36528 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
422, 12, 15, 40, 41syl31anc 1365 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑆 𝑇) ∈ (LLines‘𝐾))
4318, 31, 6, 25, 322llnmj 36576 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 𝑄) ∈ (LLines‘𝐾) ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾)))
442, 39, 42, 43syl3anc 1363 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾)))
4536, 44mpbird 258 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴)
461, 45eqeltrid 2914 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  cfv 6348  (class class class)co 7145  Basecbs 16471  joincjn 17542  meetcmee 17543  Latclat 17643  Atomscatm 36279  HLchlt 36366  LLinesclln 36507  LPlanesclpl 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515
This theorem is referenced by: (None)
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