Metamath Proof Explorer

Theorem cdlemg46

Description: Part of proof of Lemma G of Crawley p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypotheses cdlemg46.b 
|- B = ( Base ` K )
cdlemg46.h 
|- H = ( LHyp ` K )
cdlemg46.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg46.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg46 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )

Proof

Step Hyp Ref Expression
1 cdlemg46.b 
 |-  B = ( Base ` K )
2 cdlemg46.h 
 |-  H = ( LHyp ` K )
3 cdlemg46.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 cdlemg46.r 
 |-  R = ( ( trL ` K ) ` W )
5 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> K e. HL )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) )
7 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> h e. T )
8 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> h =/= ( _I |` B ) )
9 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
10 1 9 2 3 4 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ h =/= ( _I |` B ) ) -> ( R ` h ) e. ( Atoms ` K ) )
11 6 7 8 10 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` h ) e. ( Atoms ` K ) )
12 11 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` h ) e. ( Atoms ` K ) )
13 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> F e. T )
14 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> F =/= ( _I |` B ) )
15 1 9 2 3 4 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. ( Atoms ` K ) )
16 6 13 14 15 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` F ) e. ( Atoms ` K ) )
17 16 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` F ) e. ( Atoms ` K ) )
18 simpl33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` h ) =/= ( R ` F ) )
19 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` ( h o. F ) ) e. ( Atoms ` K ) )
20 2 3 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ F e. T ) -> ( h o. F ) e. T )
21 6 7 13 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( h o. F ) e. T )
22 2 3 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> `' F e. T )
23 6 13 22 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> `' F e. T )
24 eqid 
 |-  ( le ` K ) = ( le ` K )
25 eqid 
 |-  ( join ` K ) = ( join ` K )
26 24 25 2 3 4 trlco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( h o. F ) e. T /\ `' F e. T ) -> ( R ` ( ( h o. F ) o. `' F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` `' F ) ) )
27 6 21 23 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` ( ( h o. F ) o. `' F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` `' F ) ) )
28 coass 
 |-  ( ( h o. F ) o. `' F ) = ( h o. ( F o. `' F ) )
29 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : B -1-1-onto-> B )
30 6 13 29 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> F : B -1-1-onto-> B )
31 f1ococnv2 
 |-  ( F : B -1-1-onto-> B -> ( F o. `' F ) = ( _I |` B ) )
32 30 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( F o. `' F ) = ( _I |` B ) )
33 32 coeq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( h o. ( F o. `' F ) ) = ( h o. ( _I |` B ) ) )
34 1 2 3 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ h e. T ) -> h : B -1-1-onto-> B )
35 6 7 34 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> h : B -1-1-onto-> B )
36 f1of 
 |-  ( h : B -1-1-onto-> B -> h : B --> B )
37 fcoi1 
 |-  ( h : B --> B -> ( h o. ( _I |` B ) ) = h )
38 35 36 37 3syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( h o. ( _I |` B ) ) = h )
39 33 38 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( h o. ( F o. `' F ) ) = h )
40 28 39 syl5eq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( ( h o. F ) o. `' F ) = h )
41 40 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` ( ( h o. F ) o. `' F ) ) = ( R ` h ) )
42 2 3 4 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` `' F ) = ( R ` F ) )
43 6 13 42 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` `' F ) = ( R ` F ) )
44 43 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` `' F ) ) = ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
45 27 41 44 3brtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` h ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
46 45 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` h ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
47 24 25 9 hlatlej2 
 |-  ( ( K e. HL /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) /\ ( R ` F ) e. ( Atoms ` K ) ) -> ( R ` F ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
48 5 19 17 47 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` F ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
49 5 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> K e. Lat )
50 1 9 atbase 
 |-  ( ( R ` h ) e. ( Atoms ` K ) -> ( R ` h ) e. B )
51 12 50 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` h ) e. B )
52 1 9 atbase 
 |-  ( ( R ` F ) e. ( Atoms ` K ) -> ( R ` F ) e. B )
53 17 52 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` F ) e. B )
54 1 25 9 hlatjcl 
 |-  ( ( K e. HL /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) /\ ( R ` F ) e. ( Atoms ` K ) ) -> ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) e. B )
55 5 19 17 54 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) e. B )
56 1 24 25 latjle12 
 |-  ( ( K e. Lat /\ ( ( R ` h ) e. B /\ ( R ` F ) e. B /\ ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) e. B ) ) -> ( ( ( R ` h ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) /\ ( R ` F ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) ) <-> ( ( R ` h ) ( join ` K ) ( R ` F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) ) )
57 49 51 53 55 56 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( ( ( R ` h ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) /\ ( R ` F ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) ) <-> ( ( R ` h ) ( join ` K ) ( R ` F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) ) )
58 46 48 57 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( ( R ` h ) ( join ` K ) ( R ` F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) )
59 24 25 9 2atjlej 
 |-  ( ( K e. HL /\ ( ( R ` h ) e. ( Atoms ` K ) /\ ( R ` F ) e. ( Atoms ` K ) /\ ( R ` h ) =/= ( R ` F ) ) /\ ( ( R ` ( h o. F ) ) e. ( Atoms ` K ) /\ ( R ` F ) e. ( Atoms ` K ) /\ ( ( R ` h ) ( join ` K ) ( R ` F ) ) ( le ` K ) ( ( R ` ( h o. F ) ) ( join ` K ) ( R ` F ) ) ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )
60 5 12 17 18 19 17 58 59 syl133anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )
61 nelne2 
 |-  ( ( ( R ` F ) e. ( Atoms ` K ) /\ -. ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` F ) =/= ( R ` ( h o. F ) ) )
62 61 necomd 
 |-  ( ( ( R ` F ) e. ( Atoms ` K ) /\ -. ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )
63 16 62 sylan 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) /\ -. ( R ` ( h o. F ) ) e. ( Atoms ` K ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )
64 60 63 pm2.61dan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ h e. T ) /\ ( F =/= ( _I |` B ) /\ h =/= ( _I |` B ) /\ ( R ` h ) =/= ( R ` F ) ) ) -> ( R ` ( h o. F ) ) =/= ( R ` F ) )