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Theorem pmodlem2 30658
Description: Lemma for pmod1i 30659. (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 447 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  =  (/) )
21oveq1d 5889 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  (
(/)  .+  Y ) )
3 simpl1 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  K  e.  HL )
4 simpl22 1034 . . . . . 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 30623 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
( (/)  .+  Y )  =  Y )
83, 4, 7syl2anc 642 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( (/)  .+  Y
)  =  Y )
92, 8eqtrd 2328 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  Y )
109ineq1d 3382 . . 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 3410 . . . . 5  |-  ( Y 
C_  A  ->  ( Y  i^i  Z )  C_  A )
124, 11syl 15 . . . 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 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  C_  A
)
145, 6sspadd2 30627 . . . 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 1182 . . 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 3225 . 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 5882 . . . . 5  |-  ( Y  =  (/)  ->  ( X 
.+  Y )  =  ( X  .+  (/) ) )
18 simp1 955 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  K  e.  HL )
19 simp21 988 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  X  C_  A )
205, 6padd01 30622 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  .+  (/) )  =  X )
2118, 19, 20syl2anc 642 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  ( X  .+  (/) )  =  X )
2217, 21sylan9eqr 2350 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  .+  Y )  =  X )
2322ineq1d 3382 . . 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 3402 . . . 4  |-  ( X  i^i  Z )  C_  X
25 simpl1 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  K  e.  HL )
26 simpl21 1033 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  A
)
27 simpl22 1034 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  Y  C_  A
)
2827, 11syl 15 . . . . 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 30626 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3025, 26, 28, 29syl3anc 1182 . . . 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 3203 . . 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 3225 . 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 3371 . . . 4  |-  ( p  e.  ( ( X 
.+  Y )  i^i 
Z )  <->  ( p  e.  ( X  .+  Y
)  /\  p  e.  Z ) )
34 simpl1 958 . . . . . . . . . 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 30175 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
3634, 35syl 15 . . . . . . . . 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 1033 . . . . . . . . 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 1034 . . . . . . . . 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 732 . . . . . . . . 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 30611 . . . . . . . . 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 1185 . . . . . . . 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 958 . . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . . 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 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
) ) )  ->  Z  e.  S )
48 simpl3 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
) ) )  ->  X  C_  Z )
49 simpr1 961 . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . 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 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  .<_  ( q  .\/  r ) )
53 pmodlem.s . . . . . . . . . . . . . . 15  |-  S  =  ( PSubSp `  K )
5440, 41, 5, 53, 6pmodlem1 30657 . . . . . . . . . . . . . 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 1214 . . . . . . . . . . . . 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 1169 . . . . . . . . . . . 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 418 . . . . . . . . . . 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 2686 . . . . . . . . . 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 453 . . . . . . . . 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 696 . . . . . . . 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 206 . . . . . . 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 588 . . . . . 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 77 . . . . 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 573 . . . 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 208 . . 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 3198 . 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 2538 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   PSubSpcpsubsp 30307   + Pcpadd 30606
This theorem is referenced by:  pmod1i  30659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-padd 30607
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