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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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