Theorem cdleme3b 35516

Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 35523 and cdleme3 35524. (Contributed by NM, 6-Jun-2012.)

Hypotheses

Ref	Expression
cdleme1.l	|- .<_ = ( le ` K )
cdleme1.j	|- .\/ = ( join ` K )
cdleme1.m	|- ./\ = ( meet ` K )
cdleme1.a	|- A = ( Atoms ` K )
cdleme1.h	|- H = ( LHyp ` K )
cdleme1.u	|- U = ( ( P .\/ Q ) ./\ W )
cdleme1.f	|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )

Assertion

Ref	Expression
cdleme3b	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= R )

Proof of Theorem cdleme3b

Step	Hyp	Ref	Expression
1		simpll 790	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. HL )
2		simpr3l 1122	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. A )
3		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
4		cdleme1.a	. . . . 5 |- A = ( Atoms ` K )
5	3, 4	atbase 34576	. . . 4 |- ( R e. A -> R e. ( Base ` K ) )
6	2, 5	syl 17	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R e. ( Base ` K ) )
7		hllat 34650	. . . . 5 |- ( K e. HL -> K e. Lat )
8	7	ad2antrr 762	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> K e. Lat )
9		cdleme1.f	. . . . 5 |- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
10		cdleme1.l	. . . . . . . . . 10 |- .<_ = ( le ` K )
11		cdleme1.j	. . . . . . . . . 10 |- .\/ = ( join ` K )
12		cdleme1.m	. . . . . . . . . 10 |- ./\ = ( meet ` K )
13		cdleme1.h	. . . . . . . . . 10 |- H = ( LHyp ` K )
14		cdleme1.u	. . . . . . . . . 10 |- U = ( ( P .\/ Q ) ./\ W )
15	10, 11, 12, 4, 13, 14	lhpat2 35331	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
16	15	3adant3r3 1276	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. A )
17	3, 4	atbase 34576	. . . . . . . 8 |- ( U e. A -> U e. ( Base ` K ) )
18	16, 17	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U e. ( Base ` K ) )
19	3, 11	latjcl 17051	. . . . . . 7 |- ( ( K e. Lat /\ R e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( R .\/ U ) e. ( Base ` K ) )
20	8, 6, 18, 19	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ U ) e. ( Base ` K ) )
21		simpr2l 1120	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. A )
22	3, 4	atbase 34576	. . . . . . . 8 |- ( Q e. A -> Q e. ( Base ` K ) )
23	21, 22	syl 17	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> Q e. ( Base ` K ) )
24		simpr1l 1118	. . . . . . . . . 10 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. A )
25	3, 4	atbase 34576	. . . . . . . . . 10 |- ( P e. A -> P e. ( Base ` K ) )
26	24, 25	syl 17	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> P e. ( Base ` K ) )
27	3, 11	latjcl 17051	. . . . . . . . 9 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
28	8, 26, 6, 27	syl3anc 1326	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ R ) e. ( Base ` K ) )
29	3, 13	lhpbase 35284	. . . . . . . . 9 |- ( W e. H -> W e. ( Base ` K ) )
30	29	ad2antlr 763	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> W e. ( Base ` K ) )
31	3, 12	latmcl 17052	. . . . . . . 8 |- ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
32	8, 28, 30, 31	syl3anc 1326	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
33	3, 11	latjcl 17051	. . . . . . 7 |- ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
34	8, 23, 32, 33	syl3anc 1326	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) )
35	3, 12	latmcl 17052	. . . . . 6 |- ( ( K e. Lat /\ ( R .\/ U ) e. ( Base ` K ) /\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) e. ( Base ` K ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. ( Base ` K ) )
36	8, 20, 34, 35	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) e. ( Base ` K ) )
37	9, 36	syl5eqel 2705	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F e. ( Base ` K ) )
38	3, 11	latjcl 17051	. . . 4 |- ( ( K e. Lat /\ R e. ( Base ` K ) /\ F e. ( Base ` K ) ) -> ( R .\/ F ) e. ( Base ` K ) )
39	8, 6, 37, 38	syl3anc 1326	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) e. ( Base ` K ) )
40	3, 11	latjcl 17051	. . . . . . . . . 10 |- ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
41	8, 26, 23, 40	syl3anc 1326	. . . . . . . . 9 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
42	3, 10, 12	latmle2 17077	. . . . . . . . 9 |- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
43	8, 41, 30, 42	syl3anc 1326	. . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
44	14, 43	syl5eqbr 4688	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U .<_ W )
45		simpr3r 1123	. . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> -. R .<_ W )
46		nbrne2 4673	. . . . . . 7 |- ( ( U .<_ W /\ -. R .<_ W ) -> U =/= R )
47	44, 45, 46	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> U =/= R )
48	47	necomd 2849	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= U )
49		eqid 2622	. . . . . . 7 |- ( <o ` K ) = ( <o ` K )
50	11, 49, 4	atcvr1 34703	. . . . . 6 |- ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R =/= U <-> R ( <o ` K ) ( R .\/ U ) ) )
51	1, 2, 16, 50	syl3anc 1326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= U <-> R ( <o ` K ) ( R .\/ U ) ) )
52	48, 51	mpbid 222	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R ( <o ` K ) ( R .\/ U ) )
53		simpr3 1069	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R e. A /\ -. R .<_ W ) )
54	24, 21, 53	3jca 1242	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) )
55	10, 11, 12, 4, 13, 14, 9	cdleme1 35514	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
56	54, 55	syldan 487	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ F ) = ( R .\/ U ) )
57	52, 56	breqtrrd 4681	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R ( <o ` K ) ( R .\/ F ) )
58	3, 49	cvrne 34568	. . 3 |- ( ( ( K e. HL /\ R e. ( Base ` K ) /\ ( R .\/ F ) e. ( Base ` K ) ) /\ R ( <o ` K ) ( R .\/ F ) ) -> R =/= ( R .\/ F ) )
59	1, 6, 39, 57, 58	syl31anc 1329	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> R =/= ( R .\/ F ) )
60		oveq2 6658	. . . . . 6 |- ( F = R -> ( R .\/ F ) = ( R .\/ R ) )
61	60	adantl 482	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> ( R .\/ F ) = ( R .\/ R ) )
62	11, 4	hlatjidm 34655	. . . . . . 7 |- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
63	1, 2, 62	syl2anc 693	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R .\/ R ) = R )
64	63	adantr 481	. . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> ( R .\/ R ) = R )
65	61, 64	eqtr2d 2657	. . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) /\ F = R ) -> R = ( R .\/ F ) )
66	65	ex 450	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( F = R -> R = ( R .\/ F ) ) )
67	66	necon3d 2815	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> ( R =/= ( R .\/ F ) -> F =/= R ) )
68	59, 67	mpd 15	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) /\ ( R e. A /\ -. R .<_ W ) ) ) -> F =/= R )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   Latclat 17045   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LHypclh 35270

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-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-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-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274

This theorem is referenced by:  cdleme36m  35749