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Theorem pmodlem2 33491
Description: Lemma for pmod1i 33492. (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 461 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  =  (/) )
21oveq1d 6106 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  (
(/)  .+  Y ) )
3 simpl1 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  K  e.  HL )
4 simpl22 1067 . . . . . 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 33456 . . . . . 6  |-  ( ( K  e.  HL  /\  Y  C_  A )  -> 
( (/)  .+  Y )  =  Y )
83, 4, 7syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( (/)  .+  Y
)  =  Y )
92, 8eqtrd 2475 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  ( X  .+  Y )  =  Y )
109ineq1d 3551 . . 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 3578 . . . . 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 1066 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  X  =  (/) )  ->  X  C_  A
)
145, 6sspadd2 33460 . . . 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 1218 . . 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 3390 . 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 6099 . . . . 5  |-  ( Y  =  (/)  ->  ( X 
.+  Y )  =  ( X  .+  (/) ) )
18 simp1 988 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  K  e.  HL )
19 simp21 1021 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  X  C_  A )
205, 6padd01 33455 . . . . . 6  |-  ( ( K  e.  HL  /\  X  C_  A )  -> 
( X  .+  (/) )  =  X )
2118, 19, 20syl2anc 661 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  ->  ( X  .+  (/) )  =  X )
2217, 21sylan9eqr 2497 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  ( X  .+  Y )  =  X )
2322ineq1d 3551 . . 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 3570 . . . 4  |-  ( X  i^i  Z )  C_  X
25 simpl1 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  K  e.  HL )
26 simpl21 1066 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  C_  A  /\  Y  C_  A  /\  Z  e.  S )  /\  X  C_  Z )  /\  Y  =  (/) )  ->  X  C_  A
)
27 simpl22 1067 . . . . . 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 33459 . . . . 5  |-  ( ( K  e.  HL  /\  X  C_  A  /\  ( Y  i^i  Z )  C_  A )  ->  X  C_  ( X  .+  ( Y  i^i  Z ) ) )
3025, 26, 28, 29syl3anc 1218 . . . 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 3367 . . 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 3390 . 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 3539 . . . 4  |-  ( p  e.  ( ( X 
.+  Y )  i^i 
Z )  <->  ( p  e.  ( X  .+  Y
)  /\  p  e.  Z ) )
34 simpl1 991 . . . . . . . . . 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 33008 . . . . . . . . . 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 1066 . . . . . . . . 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 1067 . . . . . . . . 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 755 . . . . . . . . 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 33444 . . . . . . . . 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 1221 . . . . . . . 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 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
) ) )  ->  K  e.  HL )
45 simpl21 1066 . . . . . . . . . . . . . 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 1067 . . . . . . . . . . . . . 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 1068 . . . . . . . . . . . . . 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 993 . . . . . . . . . . . . . 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 994 . . . . . . . . . . . . . 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 1047 . . . . . . . . . . . . . 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 1048 . . . . . . . . . . . . . 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 996 . . . . . . . . . . . . . 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 33490 . . . . . . . . . . . . . 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 1250 . . . . . . . . . . . . 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 1205 . . . . . . . . . . . 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 2847 . . . . . . . . . 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 467 . . . . . . . . 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 715 . . . . . . . 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 605 . . . . . 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 590 . . . 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 3362 . 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 2690 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    i^i cin 3327    C_ wss 3328   (/)c0 3637   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   lecple 14245   joincjn 15114   Latclat 15215   Atomscatm 32908   HLchlt 32995   PSubSpcpsubsp 33140   +Pcpadd 33439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-psubsp 33147  df-padd 33440
This theorem is referenced by:  pmod1i  33492
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