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Theorem ps-2b 29671
Description: Variation of projective geometry axiom ps-2 29667. (Contributed by NM, 3-Jul-2012.)
Hypotheses
Ref Expression
ps-2b.l  |-  .<_  =  ( le `  K )
ps-2b.j  |-  .\/  =  ( join `  K )
ps-2b.m  |-  ./\  =  ( meet `  K )
ps-2b.z  |-  .0.  =  ( 0. `  K )
ps-2b.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ps-2b  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )

Proof of Theorem ps-2b
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 simp11 985 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  K  e.  HL )
2 simp12 986 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  P  e.  A )
3 simp13 987 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  Q  e.  A )
4 simp21 988 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  R  e.  A )
52, 3, 43jca 1132 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )
6 simp22 989 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  e.  A )
7 simp23 990 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  T  e.  A )
86, 7jca 518 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  e.  A  /\  T  e.  A
) )
9 simp31 991 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
10 simp32 992 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  S  =/=  T )
119, 10jca 518 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T
) )
12 simp33 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) )
13 ps-2b.l . . . 4  |-  .<_  =  ( le `  K )
14 ps-2b.j . . . 4  |-  .\/  =  ( join `  K )
15 ps-2b.a . . . 4  |-  A  =  ( Atoms `  K )
1613, 14, 15ps-2 29667 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( -.  P  .<_  ( Q  .\/  R
)  /\  S  =/=  T )  /\  ( S 
.<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
171, 5, 8, 11, 12, 16syl32anc 1190 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  ->  E. u  e.  A  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
18 simp111 1084 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  HL )
19 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
2018, 19syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  AtLat )
21 hllat 29553 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2218, 21syl 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  K  e.  Lat )
23 simp112 1085 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  P  e.  A )
24 simp121 1087 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  R  e.  A )
25 eqid 2283 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
2625, 14, 15hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
2718, 23, 24, 26syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
28 simp122 1088 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  S  e.  A )
29 simp123 1089 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  T  e.  A )
3025, 14, 15hlatjcl 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
3118, 28, 29, 30syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
32 ps-2b.m . . . . . 6  |-  ./\  =  ( meet `  K )
3325, 32latmcl 14157 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
3422, 27, 31, 33syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
) )
35 simp2 956 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  A )
36 simp3 957 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )
3725, 15atbase 29479 . . . . . . 7  |-  ( u  e.  A  ->  u  e.  ( Base `  K
) )
3835, 37syl 15 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  e.  ( Base `  K ) )
3925, 13, 32latlem12 14184 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( u  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) )  <->  u  .<_  ( ( P  .\/  R
)  ./\  ( S  .\/  T ) ) ) )
4022, 38, 27, 31, 39syl13anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  <-> 
u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) ) )
4136, 40mpbid 201 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  ->  u  .<_  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) ) )
42 ps-2b.z . . . . 5  |-  .0.  =  ( 0. `  K )
4325, 13, 42, 15atlen0 29500 . . . 4  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  R )  ./\  ( S  .\/  T ) )  e.  ( Base `  K
)  /\  u  e.  A )  /\  u  .<_  ( ( P  .\/  R )  ./\  ( S  .\/  T ) ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
4420, 34, 35, 41, 43syl31anc 1185 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/=  T  /\  ( S  .<_  ( P 
.\/  Q )  /\  T  .<_  ( Q  .\/  R ) ) ) )  /\  u  e.  A  /\  ( u  .<_  ( P 
.\/  R )  /\  u  .<_  ( S  .\/  T ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
4544rexlimdv3a 2669 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( E. u  e.  A  ( u  .<_  ( P  .\/  R )  /\  u  .<_  ( S 
.\/  T ) )  ->  ( ( P 
.\/  R )  ./\  ( S  .\/  T ) )  =/=  .0.  )
)
4617, 45mpd 14 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( -.  P  .<_  ( Q  .\/  R )  /\  S  =/= 
T  /\  ( S  .<_  ( P  .\/  Q
)  /\  T  .<_  ( Q  .\/  R ) ) ) )  -> 
( ( P  .\/  R )  ./\  ( S  .\/  T ) )  =/= 
.0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  ps-2c  29717
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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