Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmodlem2 Structured version   Unicode version

Theorem pmodlem2 33179
Description: Lemma for pmod1i 33180. (Contributed by NM, 9-Mar-2012.)
Hypotheses
Ref Expression
pmodlem.l  |-  .<_  =  ( le `  K )
pmodlem.j  |-  .\/  =  ( join `  K )
pmodlem.a  |-  A  =  ( Atoms `  K )
pmodlem.s  |-  S  =  ( PSubSp `  K )
pmodlem.p  |-  .+  =  ( +P `  K
)
Assertion
Ref Expression
pmodlem2  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
( X  .+  Y
)  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )

Proof of Theorem pmodlem2
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 458 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  =  (/) )
21oveq1d 6105 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  (
(/)  .+  Y ) )
3 simpl1 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  K  e.  HL )
4 simpl22 1062 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  Y  C_  A
)
5 pmodlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 pmodlem.p . . . . . . 7  |-  .+  =  ( +P `  K
)
75, 6padd02 33144 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
( (/)  .+  Y )  =  Y )
83, 4, 7syl2anc 656 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( (/)  .+  Y
)  =  Y )
92, 8eqtrd 2473 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  Y )
109ineq1d 3548 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( ( X 
.+  Y )  i^i 
Z )  =  ( Y  i^i  Z ) )
11 ssinss1 3575 . . . . 5  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
124, 11syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( Y  i^i  Z )  C_  A )
13 simpl21 1061 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  C_  A
)
145, 6sspadd2 33148 . . . 4  |-  ( ( K  e.  HL  /\  ( Y  i^i  Z ) 
C_  A  /\  X  C_  A )  ->  ( Y  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
153, 12, 13, 14syl3anc 1213 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( Y  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z
) ) )
1610, 15eqsstrd 3387 . 2  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( ( X 
.+  Y )  i^i 
Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
17 oveq2 6098 . . . . 5  |-  ( Y  =  (/)  ->  ( X 
.+  Y )  =  ( X  .+  (/) ) )
18 simp1 983 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  K  e.  HL )
19 simp21 1016 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  X  C_  A )
205, 6padd01 33143 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  .+  (/) )  =  X )
2118, 19, 20syl2anc 656 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  ( X  .+  (/) )  =  X )
2217, 21sylan9eqr 2495 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  .+  Y )  =  X )
2322ineq1d 3548 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( ( X 
.+  Y )  i^i 
Z )  =  ( X  i^i  Z ) )
24 inss1 3567 . . . 4  |-  ( X  i^i  Z )  C_  X
25 simpl1 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  K  e.  HL )
26 simpl21 1061 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  A
)
27 simpl22 1062 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  Y  C_  A
)
2827, 11syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( Y  i^i  Z )  C_  A )
295, 6sspadd1 33147 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3025, 26, 28, 29syl3anc 1213 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3124, 30syl5ss 3364 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z
) ) )
3223, 31eqsstrd 3387 . 2  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( ( X 
.+  Y )  i^i 
Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
33 elin 3536 . . . 4  |-  ( p  e.  ( ( X 
.+  Y )  i^i 
Z )  <->  ( p  e.  ( X  .+  Y
)  /\  p  e.  Z ) )
34 simpl1 986 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  ->  K  e.  HL )
35 hllat 32696 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
3634, 35syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  ->  K  e.  Lat )
37 simpl21 1061 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  ->  X  C_  A )
38 simpl22 1062 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  ->  Y  C_  A )
39 simprl 750 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  -> 
( X  =/=  (/)  /\  Y  =/=  (/) ) )
40 pmodlem.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
41 pmodlem.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
4240, 41, 5, 6elpaddn0 33132 . . . . . . . . 9  |-  ( ( ( K  e.  Lat  /\  X  C_  A  /\  Y  C_  A )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( p  e.  ( X  .+  Y )  <-> 
( p  e.  A  /\  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) ) ) )
4336, 37, 38, 39, 42syl31anc 1216 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  -> 
( p  e.  ( X  .+  Y )  <-> 
( p  e.  A  /\  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) ) ) )
44 simpl1 986 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  K  e.  HL )
45 simpl21 1061 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  X  C_  A )
46 simpl22 1062 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  Y  C_  A )
47 simpl23 1063 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  Z  e.  S )
48 simpl3 988 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  X  C_  Z )
49 simpr1 989 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  p  e.  Z )
50 simpr2l 1042 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  -> 
q  e.  X )
51 simpr2r 1043 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  -> 
r  e.  Y )
52 simpr3 991 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  p  .<_  ( q  .\/  r ) )
53 pmodlem.s . . . . . . . . . . . . . . 15  |-  S  =  ( PSubSp `  K )
5440, 41, 5, 53, 6pmodlem1 33178 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  X  C_  A  /\  Y  C_  A )  /\  ( Z  e.  S  /\  X  C_  Z  /\  p  e.  Z )  /\  ( q  e.  X  /\  r  e.  Y  /\  p  .<_  ( q 
.\/  r ) ) )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) )
5544, 45, 46, 47, 48, 49, 50, 51, 52, 54syl333anc 1245 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( p  e.  Z  /\  ( q  e.  X  /\  r  e.  Y )  /\  p  .<_  ( q  .\/  r
) ) )  ->  p  e.  ( X  .+  ( Y  i^i  Z
) ) )
56553exp2 1200 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
p  e.  Z  -> 
( ( q  e.  X  /\  r  e.  Y )  ->  (
p  .<_  ( q  .\/  r )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) ) ) )
5756imp 429 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  p  e.  Z
)  ->  ( (
q  e.  X  /\  r  e.  Y )  ->  ( p  .<_  ( q 
.\/  r )  ->  p  e.  ( X  .+  ( Y  i^i  Z
) ) ) ) )
5857rexlimdvv 2845 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  p  e.  Z
)  ->  ( E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r
)  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
5958adantld 464 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  p  e.  Z
)  ->  ( (
p  e.  A  /\  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r
) )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
6059adantrl 710 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  -> 
( ( p  e.  A  /\  E. q  e.  X  E. r  e.  Y  p  .<_  ( q  .\/  r ) )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
6143, 60sylbid 215 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( ( X  =/=  (/)  /\  Y  =/=  (/) )  /\  p  e.  Z ) )  -> 
( p  e.  ( X  .+  Y )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
6261exp32 602 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
( X  =/=  (/)  /\  Y  =/=  (/) )  ->  (
p  e.  Z  -> 
( p  e.  ( X  .+  Y )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) ) ) )
6362com34 83 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
( X  =/=  (/)  /\  Y  =/=  (/) )  ->  (
p  e.  ( X 
.+  Y )  -> 
( p  e.  Z  ->  p  e.  ( X 
.+  ( Y  i^i  Z ) ) ) ) ) )
6463imp4b 587 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  ( ( p  e.  ( X  .+  Y )  /\  p  e.  Z )  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
6533, 64syl5bi 217 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  ( p  e.  ( ( X  .+  Y )  i^i  Z
)  ->  p  e.  ( X  .+  ( Y  i^i  Z ) ) ) )
6665ssrdv 3359 . 2  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  ->  ( ( X 
.+  Y )  i^i 
Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
6716, 32, 66pm2.61da2ne 2688 1  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  (
( X  .+  Y
)  i^i  Z )  C_  ( X  .+  ( Y  i^i  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714    i^i cin 3324    C_ wss 3325   (/)c0 3634   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   lecple 14241   joincjn 15110   Latclat 15211   Atomscatm 32596   HLchlt 32683   PSubSpcpsubsp 32828   +Pcpadd 33127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-covers 32599  df-ats 32600  df-atl 32631  df-cvlat 32655  df-hlat 32684  df-psubsp 32835  df-padd 33128
This theorem is referenced by:  pmod1i  33180
  Copyright terms: Public domain W3C validator