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Theorem pmodlem2 30644
Description: Lemma for pmod1i 30645. (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 448 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  =  (/) )
21oveq1d 6096 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  (
(/)  .+  Y ) )
3 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  K  e.  HL )
4 simpl22 1036 . . . . . 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 30609 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
( (/)  .+  Y )  =  Y )
83, 4, 7syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( (/)  .+  Y
)  =  Y )
92, 8eqtrd 2468 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  Y )
109ineq1d 3541 . . 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 3569 . . . . 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 1035 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  C_  A
)
145, 6sspadd2 30613 . . . 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 1184 . . 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 3382 . 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 6089 . . . . 5  |-  ( Y  =  (/)  ->  ( X 
.+  Y )  =  ( X  .+  (/) ) )
18 simp1 957 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  K  e.  HL )
19 simp21 990 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  X  C_  A )
205, 6padd01 30608 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  .+  (/) )  =  X )
2118, 19, 20syl2anc 643 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  ( X  .+  (/) )  =  X )
2217, 21sylan9eqr 2490 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  .+  Y )  =  X )
2322ineq1d 3541 . . 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 3561 . . . 4  |-  ( X  i^i  Z )  C_  X
25 simpl1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  K  e.  HL )
26 simpl21 1035 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  A
)
27 simpl22 1036 . . . . . 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 30612 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3025, 26, 28, 29syl3anc 1184 . . . 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 3359 . . 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 3382 . 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 3530 . . . 4  |-  ( p  e.  ( ( X 
.+  Y )  i^i 
Z )  <->  ( p  e.  ( X  .+  Y
)  /\  p  e.  Z ) )
34 simpl1 960 . . . . . . . . . 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 30161 . . . . . . . . . 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 1035 . . . . . . . . 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 1036 . . . . . . . . 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 733 . . . . . . . . 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 30597 . . . . . . . . 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 1187 . . . . . . . 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 960 . . . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . 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 1036 . . . . . . . . . . . . . 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 1037 . . . . . . . . . . . . . 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 962 . . . . . . . . . . . . . 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 963 . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . 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 965 . . . . . . . . . . . . . 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 30643 . . . . . . . . . . . . . 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 1216 . . . . . . . . . . . . 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 1171 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 2836 . . . . . . . . . 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 454 . . . . . . . . 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 697 . . . . . . . 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 207 . . . . . . 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 589 . . . . . 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 79 . . . . 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 574 . . . 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 209 . . 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 3354 . 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 2683 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    i^i cin 3319    C_ wss 3320   (/)c0 3628   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   PSubSpcpsubsp 30293   + Pcpadd 30592
This theorem is referenced by:  pmod1i  30645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-join 14433  df-lat 14475  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-padd 30593
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