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Theorem pmodlem2 33121
Description: Lemma for pmod1i 33122. (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 462 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  =  (/) )
21oveq1d 6320 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  (
(/)  .+  Y ) )
3 simpl1 1008 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  K  e.  HL )
4 simpl22 1084 . . . . . 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 33086 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
( (/)  .+  Y )  =  Y )
83, 4, 7syl2anc 665 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( (/)  .+  Y
)  =  Y )
92, 8eqtrd 2470 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  Y )
109ineq1d 3669 . . 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 3696 . . . . 5  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
124, 11syl 17 . . . 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 1083 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  C_  A
)
145, 6sspadd2 33090 . . . 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 1264 . . 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 3504 . 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 6313 . . . . 5  |-  ( Y  =  (/)  ->  ( X 
.+  Y )  =  ( X  .+  (/) ) )
18 simp1 1005 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  K  e.  HL )
19 simp21 1038 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  X  C_  A )
205, 6padd01 33085 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  .+  (/) )  =  X )
2118, 19, 20syl2anc 665 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  ( X  .+  (/) )  =  X )
2217, 21sylan9eqr 2492 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  .+  Y )  =  X )
2322ineq1d 3669 . . 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 3688 . . . 4  |-  ( X  i^i  Z )  C_  X
25 simpl1 1008 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  K  e.  HL )
26 simpl21 1083 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  A
)
27 simpl22 1084 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  Y  C_  A
)
2827, 11syl 17 . . . . 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 33089 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3025, 26, 28, 29syl3anc 1264 . . . 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 3481 . . 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 3504 . 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 3655 . . . 4  |-  ( p  e.  ( ( X 
.+  Y )  i^i 
Z )  <->  ( p  e.  ( X  .+  Y
)  /\  p  e.  Z ) )
34 simpl1 1008 . . . . . . . . . 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 32638 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
3634, 35syl 17 . . . . . . . . 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 1083 . . . . . . . . 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 1084 . . . . . . . . 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 762 . . . . . . . . 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 33074 . . . . . . . . 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 1267 . . . . . . . 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 1008 . . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . 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 1085 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . 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 1064 . . . . . . . . . . . . . 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 1065 . . . . . . . . . . . . . 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 1013 . . . . . . . . . . . . . 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 33120 . . . . . . . . . . . . . 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 1296 . . . . . . . . . . . . 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 1223 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 2930 . . . . . . . . . 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 468 . . . . . . . . 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 720 . . . . . . . 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 218 . . . . . . 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 608 . . . . . 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 86 . . . . 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 593 . . . 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 220 . . 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 3476 . 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 2750 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    i^i cin 3441    C_ wss 3442   (/)c0 3767   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32538   HLchlt 32625   PSubSpcpsubsp 32770   +Pcpadd 33069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-psubsp 32777  df-padd 33070
This theorem is referenced by:  pmod1i  33122
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