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Theorem arglem1N 33802
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 1089 . . . . . 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 32975 . . . . . 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 1090 . . . . . 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 2462 . . . . . . 7 |- ( Base ` K ) = ( Base ` K )
7 arglem1.a . . . . . . 7 |- A = ( Atoms ` K )
86, 7atbase 32901 . . . . . 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 1091 . . . . . 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 32901 . . . . . 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 1092 . . . . . 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 32901 . . . . . 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 1093 . . . . . 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 32901 . . . . . 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 16402 . . . . 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 1285 . . . 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 467 . . . . . 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 2545 . . . . 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 1095 . . . . . . 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 2462 . . . . . . . 8 |- ( LLines ` K ) = ( LLines ` K )
2719, 7, 26llni2 33123 . . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ P =/= S ) -> ( P .\/ S ) e. ( LLines ` K ) )
282, 5, 13, 25, 27syl31anc 1279 . . . . . 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 1096 . . . . . . 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 33123 . . . . . . 7 |- ( ( ( K e. HL /\ Q e. A /\ T e. A ) /\ Q =/= T ) -> ( Q .\/ T ) e. ( LLines ` K ) )
312, 10, 16, 29, 30syl31anc 1279 . . . . . 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 2462 . . . . . . 7 |- ( LPlanes ` K ) = ( LPlanes ` K )
3419, 32, 7, 26, 332llnmj 33171 . . . . . 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 1276 . . . . 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 215 . . . 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 2540 . . 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 1094 . . . . 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 33123 . . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( LLines ` K ) )
402, 5, 10, 38, 39syl31anc 1279 . . . 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 1097 . . . . 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 33123 . . . . 5 |- ( ( ( K e. HL /\ S e. A /\ T e. A ) /\ S =/= T ) -> ( S .\/ T ) e. ( LLines ` K ) )
432, 13, 16, 41, 42syl31anc 1279 . . . 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 33171 . . . 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 1276 . . 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 240 . 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 2544 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 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   =/= wne 2633   ` cfv 5605  (class class class)co 6320   Basecbs 15176   joincjn 16244   meetcmee 16245   Latclat 16346   Atomscatm 32875   HLchlt 32962   LLinesclln 33102   LPlanesclpl 33103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-preset 16228  df-poset 16246  df-plt 16259  df-lub 16275  df-glb 16276  df-join 16277  df-meet 16278  df-p0 16340  df-lat 16347  df-clat 16409  df-oposet 32788  df-ol 32790  df-oml 32791  df-covers 32878  df-ats 32879  df-atl 32910  df-cvlat 32934  df-hlat 32963  df-llines 33109  df-lplanes 33110
This theorem is referenced by: (None)
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