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Theorem arglem1N 40173
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 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ HL)
32hllatd 39346 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ Lat)
4 simpl12 1248 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝐴)
5 eqid 2735 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
6 arglem1.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6atbase 39271 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
84, 7syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃 ∈ (Base‘𝐾))
9 simpl13 1249 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝐴)
105, 6atbase 39271 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
119, 10syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄 ∈ (Base‘𝐾))
12 simpl21 1250 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝐴)
135, 6atbase 39271 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆 ∈ (Base‘𝐾))
15 simpl22 1251 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑇𝐴)
165, 6atbase 39271 . . . . . 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 18547 . . . . 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 1378 . . . 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 484 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐺𝐴)
2321, 22eqeltrrid 2844 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
24 simpl31 1253 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑆)
25 eqid 2735 . . . . . . . 8 (LLines‘𝐾) = (LLines‘𝐾)
2618, 6, 25llni2 39495 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) ∧ 𝑃𝑆) → (𝑃 𝑆) ∈ (LLines‘𝐾))
272, 4, 12, 24, 26syl31anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑆) ∈ (LLines‘𝐾))
28 simpl32 1254 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝑇)
2918, 6, 25llni2 39495 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
302, 9, 15, 28, 29syl31anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑄 𝑇) ∈ (LLines‘𝐾))
31 arglem1.m . . . . . . 7 = (meet‘𝐾)
32 eqid 2735 . . . . . . 7 (LPlanes‘𝐾) = (LPlanes‘𝐾)
3318, 31, 6, 25, 322llnmj 39543 . . . . . 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 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾)))
3523, 34mpbid 232 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾))
3620, 35eqeltrd 2839 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾))
37 simpl23 1252 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑄)
3818, 6, 25llni2 39495 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
392, 4, 9, 37, 38syl31anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑄) ∈ (LLines‘𝐾))
40 simpl33 1255 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝑇)
4118, 6, 25llni2 39495 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
422, 12, 15, 40, 41syl31anc 1372 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑆 𝑇) ∈ (LLines‘𝐾))
4318, 31, 6, 25, 322llnmj 39543 . . . 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 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾)))
4536, 44mpbird 257 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴)
461, 45eqeltrid 2843 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  HLchlt 39332  LLinesclln 39474  LPlanesclpl 39475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482
This theorem is referenced by: (None)
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