Theorem cdlemk15 36143

Description: Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. O, D are k₁, f₁. (Contributed by NM, 1-Jul-2013.)

Hypotheses

Ref	Expression
cdlemk1.b	|- B = ( Base ` K )
cdlemk1.l	|- .<_ = ( le ` K )
cdlemk1.j	|- .\/ = ( join ` K )
cdlemk1.m	|- ./\ = ( meet ` K )
cdlemk1.a	|- A = ( Atoms ` K )
cdlemk1.h	|- H = ( LHyp ` K )
cdlemk1.t	|- T = ( ( LTrn ` K ) ` W )
cdlemk1.r	|- R = ( ( trL ` K ) ` W )
cdlemk1.s	|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
cdlemk1.o	|- O = ( S ` D )

Assertion

Ref	Expression
cdlemk15	|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) )

Distinct variable groups:   f, i, ./\   .<_ , i   .\/ , f, i   A, i   D, f, i   f, F, i   i, H   i, K   f, N, i   P, f, i   R, f, i   T, f, i   f, W, i

Allowed substitution hints:   A( f)   B( f, i)   S( f, i)   H( f)   K( f)   .<_ ( f)   O( f, i)

Proof of Theorem cdlemk15

Step	Hyp	Ref	Expression
1		simp11l 1172	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> K e. HL )
2		simp22l 1180	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> P e. A )
3		simp11 1091	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
4		simp21 1094	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> N e. T )
5		cdlemk1.l	. . . . . 6 |- .<_ = ( le ` K )
6		cdlemk1.a	. . . . . 6 |- A = ( Atoms ` K )
7		cdlemk1.h	. . . . . 6 |- H = ( LHyp ` K )
8		cdlemk1.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
9	5, 6, 7, 8	ltrnat 35426	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ P e. A ) -> ( N ` P ) e. A )
10	3, 4, 2, 9	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) e. A )
11		cdlemk1.j	. . . . 5 |- .\/ = ( join ` K )
12	5, 11, 6	hlatlej2 34662	. . . 4 |- ( ( K e. HL /\ P e. A /\ ( N ` P ) e. A ) -> ( N ` P ) .<_ ( P .\/ ( N ` P ) ) )
13	1, 2, 10, 12	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( P .\/ ( N ` P ) ) )
14		simp23 1096	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` F ) = ( R ` N ) )
15	14	oveq2d 6666	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` F ) ) = ( P .\/ ( R ` N ) ) )
16		simp22 1095	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P e. A /\ -. P .<_ W ) )
17		cdlemk1.r	. . . . . 6 |- R = ( ( trL ` K ) ` W )
18	5, 11, 6, 7, 8, 17	trljat1 35453	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ N e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
19	3, 4, 16, 18	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` N ) ) = ( P .\/ ( N ` P ) ) )
20	15, 19	eqtr2d 2657	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P .\/ ( N ` P ) ) = ( P .\/ ( R ` F ) ) )
21	13, 20	breqtrd 4679	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( P .\/ ( R ` F ) ) )
22		cdlemk1.b	. . 3 |- B = ( Base ` K )
23		cdlemk1.m	. . 3 |- ./\ = ( meet ` K )
24		cdlemk1.s	. . 3 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
25		cdlemk1.o	. . 3 |- O = ( S ` D )
26	22, 5, 11, 23, 6, 7, 8, 17, 24, 25	cdlemk14 36142	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) )
27		hllat 34650	. . . 4 |- ( K e. HL -> K e. Lat )
28	1, 27	syl 17	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> K e. Lat )
29	22, 6	atbase 34576	. . . 4 |- ( ( N ` P ) e. A -> ( N ` P ) e. B )
30	10, 29	syl 17	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) e. B )
31		simp12 1092	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> F e. T )
32		simp31 1097	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) )
33	22, 6, 7, 8, 17	trlnidat 35460	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )
34	3, 31, 32, 33	syl3anc 1326	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` F ) e. A )
35	22, 11, 6	hlatjcl 34653	. . . 4 |- ( ( K e. HL /\ P e. A /\ ( R ` F ) e. A ) -> ( P .\/ ( R ` F ) ) e. B )
36	1, 2, 34, 35	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( P .\/ ( R ` F ) ) e. B )
37	25	fveq1i 6192	. . . . 5 |- ( O ` P ) = ( ( S ` D ) ` P )
38	22, 5, 11, 6, 7, 8, 17, 23, 24	cdlemksat 36134	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( S ` D ) ` P ) e. A )
39	37, 38	syl5eqel 2705	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( O ` P ) e. A )
40		simp13 1093	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> D e. T )
41		simp33 1099	. . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` D ) =/= ( R ` F ) )
42	41	necomd 2849	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` F ) =/= ( R ` D ) )
43	6, 7, 8, 17	trlcocnvat 36012	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T ) /\ ( R ` F ) =/= ( R ` D ) ) -> ( R ` ( F o. `' D ) ) e. A )
44	3, 31, 40, 42, 43	syl121anc 1331	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( R ` ( F o. `' D ) ) e. A )
45	22, 11, 6	hlatjcl 34653	. . . 4 |- ( ( K e. HL /\ ( O ` P ) e. A /\ ( R ` ( F o. `' D ) ) e. A ) -> ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) e. B )
46	1, 39, 44, 45	syl3anc 1326	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) e. B )
47	22, 5, 23	latlem12 17078	. . 3 |- ( ( K e. Lat /\ ( ( N ` P ) e. B /\ ( P .\/ ( R ` F ) ) e. B /\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) e. B ) ) -> ( ( ( N ` P ) .<_ ( P .\/ ( R ` F ) ) /\ ( N ` P ) .<_ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) <-> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) ) )
48	28, 30, 36, 46, 47	syl13anc 1328	. 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( ( ( N ` P ) .<_ ( P .\/ ( R ` F ) ) /\ ( N ` P ) .<_ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) <-> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) ) )
49	21, 26, 48	mpbi2and 956	1 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ ( R ` D ) =/= ( R ` F ) ) ) -> ( N ` P ) .<_ ( ( P .\/ ( R ` F ) ) ./\ ( ( O ` P ) .\/ ( R ` ( F o. `' D ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-iin 4523  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-mpt2 6655  df-1st 7168  df-2nd 7169  df-undef 7399  df-map 7859  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-p1 17040  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  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446

This theorem is referenced by:  cdlemk17  36146  cdlemk15-2N  36167