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Metamath Proof Explorer

Theorem dalem25

Description: Lemma for dath . Show that the dummy center of perspectivity c is different from auxiliary atom G . (Contributed by NM, 3-Aug-2012)

Ref Expression
Hypotheses dalem.ph 
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalem.l 
|- .<_ = ( le ` K )
dalem.j 
|- .\/ = ( join ` K )
dalem.a 
|- A = ( Atoms ` K )
dalem.ps 
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
dalem23.m 
|- ./\ = ( meet ` K )
dalem23.o 
|- O = ( LPlanes ` K )
dalem23.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem23.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem23.g 
|- G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
Assertion dalem25 
|- ( ( ph /\ Y = Z /\ ps ) -> c =/= G )

Proof

Step Hyp Ref Expression
1 dalem.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2 dalem.l 
 |-  .<_ = ( le ` K )
3 dalem.j 
 |-  .\/ = ( join ` K )
4 dalem.a 
 |-  A = ( Atoms ` K )
5 dalem.ps 
 |-  ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
6 dalem23.m 
 |-  ./\ = ( meet ` K )
7 dalem23.o 
 |-  O = ( LPlanes ` K )
8 dalem23.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem23.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem23.g 
 |-  G = ( ( c .\/ P ) ./\ ( d .\/ S ) )
11 1 2 3 4 dalemcnes 
 |-  ( ph -> C =/= S )
12 11 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C =/= S )
13 5 dalemclccjdd 
 |-  ( ps -> C .<_ ( c .\/ d ) )
14 13 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( c .\/ d ) )
15 14 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> C .<_ ( c .\/ d ) )
16 simpr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> c = G )
17 1 dalemkelat 
 |-  ( ph -> K e. Lat )
18 17 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. Lat )
19 1 dalemkehl 
 |-  ( ph -> K e. HL )
20 19 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
21 5 dalemccea 
 |-  ( ps -> c e. A )
22 21 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. A )
23 1 dalempea 
 |-  ( ph -> P e. A )
24 23 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> P e. A )
25 eqid 
 |-  ( Base ` K ) = ( Base ` K )
26 25 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ P e. A ) -> ( c .\/ P ) e. ( Base ` K ) )
27 20 22 24 26 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ P ) e. ( Base ` K ) )
28 5 dalemddea 
 |-  ( ps -> d e. A )
29 28 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. A )
30 1 dalemsea 
 |-  ( ph -> S e. A )
31 30 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S e. A )
32 25 3 4 hlatjcl 
 |-  ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) e. ( Base ` K ) )
33 20 29 31 32 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) e. ( Base ` K ) )
34 25 2 6 latmle2 
 |-  ( ( K e. Lat /\ ( c .\/ P ) e. ( Base ` K ) /\ ( d .\/ S ) e. ( Base ` K ) ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) .<_ ( d .\/ S ) )
35 18 27 33 34 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .\/ P ) ./\ ( d .\/ S ) ) .<_ ( d .\/ S ) )
36 10 35 eqbrtrid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G .<_ ( d .\/ S ) )
37 3 4 hlatjcom 
 |-  ( ( K e. HL /\ d e. A /\ S e. A ) -> ( d .\/ S ) = ( S .\/ d ) )
38 20 29 31 37 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( d .\/ S ) = ( S .\/ d ) )
39 36 38 breqtrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> G .<_ ( S .\/ d ) )
40 39 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> G .<_ ( S .\/ d ) )
41 16 40 eqbrtrd 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> c .<_ ( S .\/ d ) )
42 2 3 4 hlatlej2 
 |-  ( ( K e. HL /\ S e. A /\ d e. A ) -> d .<_ ( S .\/ d ) )
43 20 31 29 42 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d .<_ ( S .\/ d ) )
44 43 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> d .<_ ( S .\/ d ) )
45 5 4 dalemcceb 
 |-  ( ps -> c e. ( Base ` K ) )
46 45 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c e. ( Base ` K ) )
47 25 4 atbase 
 |-  ( d e. A -> d e. ( Base ` K ) )
48 28 47 syl 
 |-  ( ps -> d e. ( Base ` K ) )
49 48 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> d e. ( Base ` K ) )
50 25 3 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ d e. A ) -> ( S .\/ d ) e. ( Base ` K ) )
51 20 31 29 50 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( S .\/ d ) e. ( Base ` K ) )
52 25 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( c e. ( Base ` K ) /\ d e. ( Base ` K ) /\ ( S .\/ d ) e. ( Base ` K ) ) ) -> ( ( c .<_ ( S .\/ d ) /\ d .<_ ( S .\/ d ) ) <-> ( c .\/ d ) .<_ ( S .\/ d ) ) )
53 18 46 49 51 52 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( c .<_ ( S .\/ d ) /\ d .<_ ( S .\/ d ) ) <-> ( c .\/ d ) .<_ ( S .\/ d ) ) )
54 53 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( ( c .<_ ( S .\/ d ) /\ d .<_ ( S .\/ d ) ) <-> ( c .\/ d ) .<_ ( S .\/ d ) ) )
55 41 44 54 mpbi2and 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( c .\/ d ) .<_ ( S .\/ d ) )
56 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
57 56 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C e. ( Base ` K ) )
58 25 3 4 hlatjcl 
 |-  ( ( K e. HL /\ c e. A /\ d e. A ) -> ( c .\/ d ) e. ( Base ` K ) )
59 20 22 29 58 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) e. ( Base ` K ) )
60 25 2 lattr 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( c .\/ d ) e. ( Base ` K ) /\ ( S .\/ d ) e. ( Base ` K ) ) ) -> ( ( C .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ ( S .\/ d ) ) -> C .<_ ( S .\/ d ) ) )
61 18 57 59 51 60 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( C .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ ( S .\/ d ) ) -> C .<_ ( S .\/ d ) ) )
62 61 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( ( C .<_ ( c .\/ d ) /\ ( c .\/ d ) .<_ ( S .\/ d ) ) -> C .<_ ( S .\/ d ) ) )
63 15 55 62 mp2and 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> C .<_ ( S .\/ d ) )
64 1 7 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
65 64 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> Y e. ( Base ` K ) )
66 25 2 6 latmlem1 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( S .\/ d ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( C .<_ ( S .\/ d ) -> ( C ./\ Y ) .<_ ( ( S .\/ d ) ./\ Y ) ) )
67 18 57 51 65 66 syl13anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( C .<_ ( S .\/ d ) -> ( C ./\ Y ) .<_ ( ( S .\/ d ) ./\ Y ) ) )
68 67 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( C .<_ ( S .\/ d ) -> ( C ./\ Y ) .<_ ( ( S .\/ d ) ./\ Y ) ) )
69 63 68 mpd 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( C ./\ Y ) .<_ ( ( S .\/ d ) ./\ Y ) )
70 1 2 3 4 7 8 9 dalem17 
 |-  ( ( ph /\ Y = Z ) -> C .<_ Y )
71 70 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> C .<_ Y )
72 25 2 6 latleeqm1 
 |-  ( ( K e. Lat /\ C e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( C .<_ Y <-> ( C ./\ Y ) = C ) )
73 18 57 65 72 syl3anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( C .<_ Y <-> ( C ./\ Y ) = C ) )
74 71 73 mpbid 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( C ./\ Y ) = C )
75 74 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( C ./\ Y ) = C )
76 1 2 3 4 9 dalemsly 
 |-  ( ( ph /\ Y = Z ) -> S .<_ Y )
77 76 3adant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> S .<_ Y )
78 5 dalem-ddly 
 |-  ( ps -> -. d .<_ Y )
79 78 3ad2ant3 
 |-  ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
80 25 2 3 6 4 2atjm 
 |-  ( ( K e. HL /\ ( S e. A /\ d e. A /\ Y e. ( Base ` K ) ) /\ ( S .<_ Y /\ -. d .<_ Y ) ) -> ( ( S .\/ d ) ./\ Y ) = S )
81 20 31 29 65 77 79 80 syl132anc 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( ( S .\/ d ) ./\ Y ) = S )
82 81 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( ( S .\/ d ) ./\ Y ) = S )
83 69 75 82 3brtr3d 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> C .<_ S )
84 hlatl 
 |-  ( K e. HL -> K e. AtLat )
85 19 84 syl 
 |-  ( ph -> K e. AtLat )
86 1 2 3 4 7 8 dalemcea 
 |-  ( ph -> C e. A )
87 2 4 atcmp 
 |-  ( ( K e. AtLat /\ C e. A /\ S e. A ) -> ( C .<_ S <-> C = S ) )
88 85 86 30 87 syl3anc 
 |-  ( ph -> ( C .<_ S <-> C = S ) )
89 88 3ad2ant1 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( C .<_ S <-> C = S ) )
90 89 adantr 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> ( C .<_ S <-> C = S ) )
91 83 90 mpbid 
 |-  ( ( ( ph /\ Y = Z /\ ps ) /\ c = G ) -> C = S )
92 91 ex 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( c = G -> C = S ) )
93 92 necon3d 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( C =/= S -> c =/= G ) )
94 12 93 mpd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> c =/= G )