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Theorem arglem1N 35477
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 e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> F e. A )
Proof of Theorem arglem1N
StepHypRef Expression
1 arglem1.f . 2 |- F = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
2 simpl11 1136 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> K e. HL )
3 hllat 34650 . . . . . 6 |- ( K e. HL -> K e. Lat )
42, 3syl 17 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> K e. Lat )
5 simpl12 1137 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> P e. A )
6 eqid 2622 . . . . . . 7 |- ( Base ` K ) = ( Base ` K )
7 arglem1.a . . . . . . 7 |- A = ( Atoms ` K )
86, 7atbase 34576 . . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
95, 8syl 17 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> P e. ( Base ` K ) )
10 simpl13 1138 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> Q e. A )
116, 7atbase 34576 . . . . . 6 |- ( Q e. A -> Q e. ( Base ` K ) )
1210, 11syl 17 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> Q e. ( Base ` K ) )
13 simpl21 1139 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> S e. A )
146, 7atbase 34576 . . . . . 6 |- ( S e. A -> S e. ( Base ` K ) )
1513, 14syl 17 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> S e. ( Base ` K ) )
16 simpl22 1140 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> T e. A )
176, 7atbase 34576 . . . . . 6 |- ( T e. A -> T e. ( Base ` K ) )
1816, 17syl 17 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> T e. ( Base ` K ) )
19 arglem1.j . . . . . 6 |- .\/ = ( join ` K )
206, 19latj4 17101 . . . . 5 |- ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) )
214, 9, 12, 15, 18, 20syl122anc 1335 . . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) )
22 arglem1.g . . . . . 6 |- G = ( ( P .\/ S ) ./\ ( Q .\/ T ) )
23 simpr 477 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> G e. A )
2422, 23syl5eqelr 2706 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A )
25 simpl31 1142 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> P =/= S )
26 eqid 2622 . . . . . . . 8 |- ( LLines ` K ) = ( LLines ` K )
2719, 7, 26llni2 34798 . . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ P =/= S ) -> ( P .\/ S ) e. ( LLines ` K ) )
282, 5, 13, 25, 27syl31anc 1329 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( P .\/ S ) e. ( LLines ` K ) )
29 simpl32 1143 . . . . . . 7 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> Q =/= T )
3019, 7, 26llni2 34798 . . . . . . 7 |- ( ( ( K e. HL /\ Q e. A /\ T e. A ) /\ Q =/= T ) -> ( Q .\/ T ) e. ( LLines ` K ) )
312, 10, 16, 29, 30syl31anc 1329 . . . . . 6 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( Q .\/ T ) e. ( LLines ` K ) )
32 arglem1.m . . . . . . 7 |- ./\ = ( meet ` K )
33 eqid 2622 . . . . . . 7 |- ( LPlanes ` K ) = ( LPlanes ` K )
3419, 32, 7, 26, 332llnmj 34846 . . . . . 6 |- ( ( K e. HL /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. ( LPlanes ` K ) ) )
352, 28, 31, 34syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( ( P .\/ S ) ./\ ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. ( LPlanes ` K ) ) )
3624, 35mpbid 222 . . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. ( LPlanes ` K ) )
3721, 36eqeltrd 2701 . . 3 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. ( LPlanes ` K ) )
38 simpl23 1141 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> P =/= Q )
3919, 7, 26llni2 34798 . . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( LLines ` K ) )
402, 5, 10, 38, 39syl31anc 1329 . . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( P .\/ Q ) e. ( LLines ` K ) )
41 simpl33 1144 . . . . 5 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> S =/= T )
4219, 7, 26llni2 34798 . . . . 5 |- ( ( ( K e. HL /\ S e. A /\ T e. A ) /\ S =/= T ) -> ( S .\/ T ) e. ( LLines ` K ) )
432, 13, 16, 41, 42syl31anc 1329 . . . 4 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( S .\/ T ) e. ( LLines ` K ) )
4419, 32, 7, 26, 332llnmj 34846 . . . 4 |- ( ( K e. HL /\ ( P .\/ Q ) e. ( LLines ` K ) /\ ( S .\/ T ) e. ( LLines ` K ) ) -> ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A <-> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. ( LPlanes ` K ) ) )
452, 40, 43, 44syl3anc 1326 . . 3 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A <-> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. ( LPlanes ` K ) ) )
4637, 45mpbird 247 . 2 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A )
471, 46syl5eqel 2705 1 |- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A /\ P =/= Q ) /\ ( P =/= S /\ Q =/= T /\ S =/= T ) ) /\ G e. A ) -> F e. A )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   HLchlt 34637   LLinesclln 34777   LPlanesclpl 34778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785
This theorem is referenced by: (None)
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