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Theorem dihjustlem 30685
Description: Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
Hypotheses
Ref Expression
dihjust.b  |-  B  =  ( Base `  K
)
dihjust.l  |-  .<_  =  ( le `  K )
dihjust.j  |-  .\/  =  ( join `  K )
dihjust.m  |-  ./\  =  ( meet `  K )
dihjust.a  |-  A  =  ( Atoms `  K )
dihjust.h  |-  H  =  ( LHyp `  K
)
dihjust.i  |-  I  =  ( ( DIsoB `  K
) `  W )
dihjust.J  |-  J  =  ( ( DIsoC `  K
) `  W )
dihjust.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihjust.s  |-  .(+)  =  (
LSSum `  U )
Assertion
Ref Expression
dihjustlem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )

Proof of Theorem dihjustlem
StepHypRef Expression
1 simp1l 979 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  K  e.  HL )
2 hllat 28832 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  K  e.  Lat )
4 simp21l 1072 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  Q  e.  A )
5 dihjust.b . . . . . . 7  |-  B  =  ( Base `  K
)
6 dihjust.a . . . . . . 7  |-  A  =  ( Atoms `  K )
75, 6atbase 28758 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
84, 7syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  Q  e.  B )
9 simp23 990 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  X  e.  B )
10 simp1r 980 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  W  e.  H )
11 dihjust.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
125, 11lhpbase 29466 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  B )
1310, 12syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  W  e.  B )
14 dihjust.m . . . . . . 7  |-  ./\  =  ( meet `  K )
155, 14latmcl 14153 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
163, 9, 13, 15syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( X  ./\ 
W )  e.  B
)
17 dihjust.l . . . . . 6  |-  .<_  =  ( le `  K )
18 dihjust.j . . . . . 6  |-  .\/  =  ( join `  K )
195, 17, 18latlej1 14162 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  B  /\  ( X  ./\  W )  e.  B )  ->  Q  .<_  ( Q  .\/  ( X  ./\  W ) ) )
203, 8, 16, 19syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  Q  .<_  ( Q  .\/  ( X 
./\  W ) ) )
21 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )
2220, 21breqtrd 4048 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  Q  .<_  ( R  .\/  ( X 
./\  W ) ) )
23 simp1 955 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
24 simp22 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( R  e.  A  /\  -.  R  .<_  W ) )
25 simp21 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
265, 17, 14latmle2 14179 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  .<_  W )
273, 9, 13, 26syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( X  ./\ 
W )  .<_  W )
2816, 27jca 518 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( ( X  ./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )
29 dihjust.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
30 dihjust.J . . . . 5  |-  J  =  ( ( DIsoC `  K
) `  W )
31 dihjust.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
32 dihjust.s . . . . 5  |-  .(+)  =  (
LSSum `  U )
335, 17, 18, 6, 11, 29, 30, 31, 32cdlemn 30681 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) ) )  ->  ( Q  .<_  ( R  .\/  ( X 
./\  W ) )  <-> 
( J `  Q
)  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) ) )
3423, 24, 25, 28, 33syl13anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( Q  .<_  ( R  .\/  ( X  ./\  W ) )  <-> 
( J `  Q
)  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) ) )
3522, 34mpbid 201 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( J `  Q )  C_  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) ) )
3611, 31, 23dvhlmod 30579 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  U  e.  LMod )
37 eqid 2284 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
3837lsssssubg 15711 . . . . 5  |-  ( U  e.  LMod  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
3936, 38syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( LSubSp `  U )  C_  (SubGrp `  U ) )
4017, 6, 11, 31, 30, 37diclss 30662 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  -> 
( J `  R
)  e.  ( LSubSp `  U ) )
4123, 24, 40syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( J `  R )  e.  (
LSubSp `  U ) )
4239, 41sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( J `  R )  e.  (SubGrp `  U ) )
435, 17, 11, 31, 29, 37diblss 30639 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X 
./\  W )  e.  B  /\  ( X 
./\  W )  .<_  W ) )  -> 
( I `  ( X  ./\  W ) )  e.  ( LSubSp `  U
) )
4423, 16, 27, 43syl12anc 1180 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( I `  ( X  ./\  W
) )  e.  (
LSubSp `  U ) )
4539, 44sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( I `  ( X  ./\  W
) )  e.  (SubGrp `  U ) )
4632lsmub2 14964 . . 3  |-  ( ( ( J `  R
)  e.  (SubGrp `  U )  /\  (
I `  ( X  ./\ 
W ) )  e.  (SubGrp `  U )
)  ->  ( I `  ( X  ./\  W
) )  C_  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) ) )
4742, 45, 46syl2anc 642 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( I `  ( X  ./\  W
) )  C_  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) ) )
4817, 6, 11, 31, 30, 37diclss 30662 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( J `  Q
)  e.  ( LSubSp `  U ) )
4923, 25, 48syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( J `  Q )  e.  (
LSubSp `  U ) )
5039, 49sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( J `  Q )  e.  (SubGrp `  U ) )
5137, 32lsmcl 15832 . . . . 5  |-  ( ( U  e.  LMod  /\  ( J `  R )  e.  ( LSubSp `  U )  /\  ( I `  ( X  ./\  W ) )  e.  ( LSubSp `  U
) )  ->  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) )  e.  (
LSubSp `  U ) )
5236, 41, 44, 51syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) )  e.  ( LSubSp `  U ) )
5339, 52sseldd 3182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) )  e.  (SubGrp `  U ) )
5432lsmlub 14970 . . 3  |-  ( ( ( J `  Q
)  e.  (SubGrp `  U )  /\  (
I `  ( X  ./\ 
W ) )  e.  (SubGrp `  U )  /\  ( ( J `  R )  .(+)  ( I `
 ( X  ./\  W ) ) )  e.  (SubGrp `  U )
)  ->  ( (
( J `  Q
)  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) )  /\  ( I `
 ( X  ./\  W ) )  C_  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) ) )  <->  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) ) )
5550, 45, 53, 54syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( (
( J `  Q
)  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) )  /\  ( I `
 ( X  ./\  W ) )  C_  (
( J `  R
)  .(+)  ( I `  ( X  ./\  W ) ) ) )  <->  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) ) )
5635, 47, 55mpbi2and 887 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W )  /\  X  e.  B
)  /\  ( Q  .\/  ( X  ./\  W
) )  =  ( R  .\/  ( X 
./\  W ) ) )  ->  ( ( J `  Q )  .(+)  ( I `  ( X  ./\  W ) ) )  C_  ( ( J `  R )  .(+)  ( I `  ( X  ./\  W ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685    C_ wss 3153   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   joincjn 14074   meetcmee 14075   Latclat 14147  SubGrpcsubg 14611   LSSumclsm 14941   LModclmod 15623   LSubSpclss 15685   Atomscatm 28732   HLchlt 28819   LHypclh 29452   DVecHcdvh 30547   DIsoBcdib 30607   DIsoCcdic 30641
This theorem is referenced by:  dihjust  30686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10779  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-sca 13220  df-vsca 13221  df-0g 13400  df-poset 14076  df-plt 14088  df-lub 14104  df-glb 14105  df-join 14106  df-meet 14107  df-p0 14141  df-p1 14142  df-lat 14148  df-clat 14210  df-mnd 14363  df-submnd 14412  df-grp 14485  df-minusg 14486  df-sbg 14487  df-subg 14614  df-cntz 14789  df-lsm 14943  df-cmn 15087  df-abl 15088  df-mgp 15322  df-rng 15336  df-ur 15338  df-oppr 15401  df-dvdsr 15419  df-unit 15420  df-invr 15450  df-dvr 15461  df-drng 15510  df-lmod 15625  df-lss 15686  df-lsp 15725  df-lvec 15852  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-llines 28966  df-lplanes 28967  df-lvols 28968  df-lines 28969  df-psubsp 28971  df-pmap 28972  df-padd 29264  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573  df-trl 29627  df-tendo 30223  df-edring 30225  df-disoa 30498  df-dvech 30548  df-dib 30608  df-dic 30642
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