Theorem arglem1N 35203

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

Step	Hyp	Ref	Expression
1		arglem1.f	. 2 |- F = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
2		simpl11 1071	. . . . . 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 34377	. . . . . 6 |- ( K e. HL -> K e. Lat )
4	2, 3	syl 16	. . . . 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 1072	. . . . . 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 2467	. . . . . . 7 |- ( Base ` K ) = ( Base ` K )
7		arglem1.a	. . . . . . 7 |- A = ( Atoms ` K )
8	6, 7	atbase 34303	. . . . . 6 |- ( P e. A -> P e. ( Base ` K ) )
9	5, 8	syl 16	. . . . 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 1073	. . . . . 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 )
11	6, 7	atbase 34303	. . . . . 6 |- ( Q e. A -> Q e. ( Base ` K ) )
12	10, 11	syl 16	. . . . 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 1074	. . . . . 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 )
14	6, 7	atbase 34303	. . . . . 6 |- ( S e. A -> S e. ( Base ` K ) )
15	13, 14	syl 16	. . . . 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 1075	. . . . . 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 )
17	6, 7	atbase 34303	. . . . . 6 |- ( T e. A -> T e. ( Base ` K ) )
18	16, 17	syl 16	. . . . 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 )
20	6, 19	latj4 15591	. . . . 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 ) ) )
21	4, 9, 12, 15, 18, 20	syl122anc 1237	. . . 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 461	. . . . . 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 )
24	22, 23	syl5eqelr 2560	. . . . 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 1077	. . . . . . 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 2467	. . . . . . . 8 |- ( LLines ` K ) = ( LLines ` K )
27	19, 7, 26	llni2 34525	. . . . . . 7 |- ( ( ( K e. HL /\ P e. A /\ S e. A ) /\ P =/= S ) -> ( P .\/ S ) e. ( LLines ` K ) )
28	2, 5, 13, 25, 27	syl31anc 1231	. . . . . 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 1078	. . . . . . 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 )
30	19, 7, 26	llni2 34525	. . . . . . 7 |- ( ( ( K e. HL /\ Q e. A /\ T e. A ) /\ Q =/= T ) -> ( Q .\/ T ) e. ( LLines ` K ) )
31	2, 10, 16, 29, 30	syl31anc 1231	. . . . . 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 2467	. . . . . . 7 |- ( LPlanes ` K ) = ( LPlanes ` K )
34	19, 32, 7, 26, 33	2llnmj 34573	. . . . . 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 ) ) )
35	2, 28, 31, 34	syl3anc 1228	. . . . 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 ) ) )
36	24, 35	mpbid 210	. . . 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 ) )
37	21, 36	eqeltrd 2555	. . 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 1076	. . . . 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 )
39	19, 7, 26	llni2 34525	. . . . 5 |- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( LLines ` K ) )
40	2, 5, 10, 38, 39	syl31anc 1231	. . . 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 1079	. . . . 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 )
42	19, 7, 26	llni2 34525	. . . . 5 |- ( ( ( K e. HL /\ S e. A /\ T e. A ) /\ S =/= T ) -> ( S .\/ T ) e. ( LLines ` K ) )
43	2, 13, 16, 41, 42	syl31anc 1231	. . . 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 ) )
44	19, 32, 7, 26, 33	2llnmj 34573	. . . 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 ) ) )
45	2, 40, 43, 44	syl3anc 1228	. . 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 ) ) )
46	37, 45	mpbird 232	. 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 )
47	1, 46	syl5eqel 2559	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 )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   ` cfv 5588  (class class class)co 6285   Basecbs 14493   joincjn 15434   meetcmee 15435   Latclat 15535   Atomscatm 34277   HLchlt 34364   LLinesclln 34504   LPlanesclpl 34505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512

This theorem is referenced by: (None)