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Theorem arglem1N 34495
 Description: Lemma for Desargues' law. Theorem 13.3 of [Crawley] p. 110, 3rd and 4th 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 1129 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ HL)
3 hllat 33668 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
42, 3syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐾 ∈ Lat)
5 simpl12 1130 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝐴)
6 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 arglem1.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7atbase 33594 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
95, 8syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃 ∈ (Base‘𝐾))
10 simpl13 1131 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝐴)
116, 7atbase 33594 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄 ∈ (Base‘𝐾))
13 simpl21 1132 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝐴)
146, 7atbase 33594 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
1513, 14syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆 ∈ (Base‘𝐾))
16 simpl22 1133 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑇𝐴)
176, 7atbase 33594 . . . . . 6 (𝑇𝐴𝑇 ∈ (Base‘𝐾))
1816, 17syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑇 ∈ (Base‘𝐾))
19 arglem1.j . . . . . 6 = (join‘𝐾)
206, 19latj4 16924 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾))) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
214, 9, 12, 15, 18, 20syl122anc 1327 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) = ((𝑃 𝑆) (𝑄 𝑇)))
22 arglem1.g . . . . . 6 𝐺 = ((𝑃 𝑆) (𝑄 𝑇))
23 simpr 476 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐺𝐴)
2422, 23syl5eqelr 2693 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴)
25 simpl31 1135 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑆)
26 eqid 2610 . . . . . . . 8 (LLines‘𝐾) = (LLines‘𝐾)
2719, 7, 26llni2 33816 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) ∧ 𝑃𝑆) → (𝑃 𝑆) ∈ (LLines‘𝐾))
282, 5, 13, 25, 27syl31anc 1321 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑆) ∈ (LLines‘𝐾))
29 simpl32 1136 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑄𝑇)
3019, 7, 26llni2 33816 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑄𝐴𝑇𝐴) ∧ 𝑄𝑇) → (𝑄 𝑇) ∈ (LLines‘𝐾))
312, 10, 16, 29, 30syl31anc 1321 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑄 𝑇) ∈ (LLines‘𝐾))
32 arglem1.m . . . . . . 7 = (meet‘𝐾)
33 eqid 2610 . . . . . . 7 (LPlanes‘𝐾) = (LPlanes‘𝐾)
3419, 32, 7, 26, 332llnmj 33864 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 𝑆) ∈ (LLines‘𝐾) ∧ (𝑄 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾)))
352, 28, 31, 34syl3anc 1318 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑆) (𝑄 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾)))
3624, 35mpbid 221 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑆) (𝑄 𝑇)) ∈ (LPlanes‘𝐾))
3721, 36eqeltrd 2688 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾))
38 simpl23 1134 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑃𝑄)
3919, 7, 26llni2 33816 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑃 𝑄) ∈ (LLines‘𝐾))
402, 5, 10, 38, 39syl31anc 1321 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑃 𝑄) ∈ (LLines‘𝐾))
41 simpl33 1137 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝑆𝑇)
4219, 7, 26llni2 33816 . . . . 5 (((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 𝑇) ∈ (LLines‘𝐾))
432, 13, 16, 41, 42syl31anc 1321 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (𝑆 𝑇) ∈ (LLines‘𝐾))
4419, 32, 7, 26, 332llnmj 33864 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 𝑄) ∈ (LLines‘𝐾) ∧ (𝑆 𝑇) ∈ (LLines‘𝐾)) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾)))
452, 40, 43, 44syl3anc 1318 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → (((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴 ↔ ((𝑃 𝑄) (𝑆 𝑇)) ∈ (LPlanes‘𝐾)))
4637, 45mpbird 246 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → ((𝑃 𝑄) (𝑆 𝑇)) ∈ 𝐴)
471, 46syl5eqel 2692 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  HLchlt 33655  LLinesclln 33795  LPlanesclpl 33796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803 This theorem is referenced by: (None)
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