HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslmd1i Unicode version

Theorem mdslmd1i 22964
Description: Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (meet version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslmd.1  |-  A  e. 
CH
mdslmd.2  |-  B  e. 
CH
mdslmd.3  |-  C  e. 
CH
mdslmd.4  |-  D  e. 
CH
Assertion
Ref Expression
mdslmd1i  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( C  MH  D  <->  ( C  i^i  B )  MH  ( D  i^i  B ) ) )

Proof of Theorem mdslmd1i
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssin 3425 . . 3  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
2 mdslmd.3 . . . 4  |-  C  e. 
CH
3 mdslmd.4 . . . 4  |-  D  e. 
CH
4 mdslmd.1 . . . . 5  |-  A  e. 
CH
5 mdslmd.2 . . . . 5  |-  B  e. 
CH
64, 5chjcli 22091 . . . 4  |-  ( A  vH  B )  e. 
CH
72, 3, 6chlubi 22105 . . 3  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
81, 7anbi12i 678 . 2  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  <->  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )
9 chjcl 21991 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  A  e.  CH )  ->  ( x  vH  A
)  e.  CH )
104, 9mpan2 652 . . . . . . . . . 10  |-  ( x  e.  CH  ->  (
x  vH  A )  e.  CH )
11 sseq1 3233 . . . . . . . . . . . 12  |-  ( y  =  ( x  vH  A )  ->  (
y  C_  D  <->  ( x  vH  A )  C_  D
) )
12 oveq1 5907 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  C )  =  ( ( x  vH  A )  vH  C ) )
1312ineq1d 3403 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
( y  vH  C
)  i^i  D )  =  ( ( ( x  vH  A )  vH  C )  i^i 
D ) )
14 oveq1 5907 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  ( C  i^i  D ) )  =  ( ( x  vH  A )  vH  ( C  i^i  D ) ) )
1513, 14sseq12d 3241 . . . . . . . . . . . 12  |-  ( y  =  ( x  vH  A )  ->  (
( ( y  vH  C )  i^i  D
)  C_  ( y  vH  ( C  i^i  D
) )  <->  ( (
( x  vH  A
)  vH  C )  i^i  D )  C_  (
( x  vH  A
)  vH  ( C  i^i  D ) ) ) )
1611, 15imbi12d 311 . . . . . . . . . . 11  |-  ( y  =  ( x  vH  A )  ->  (
( y  C_  D  ->  ( ( y  vH  C )  i^i  D
)  C_  ( y  vH  ( C  i^i  D
) ) )  <->  ( (
x  vH  A )  C_  D  ->  ( (
( x  vH  A
)  vH  C )  i^i  D )  C_  (
( x  vH  A
)  vH  ( C  i^i  D ) ) ) ) )
1716rspcv 2914 . . . . . . . . . 10  |-  ( ( x  vH  A )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  D  ->  ( ( y  vH  C
)  i^i  D )  C_  ( y  vH  ( C  i^i  D ) ) )  ->  ( (
x  vH  A )  C_  D  ->  ( (
( x  vH  A
)  vH  C )  i^i  D )  C_  (
( x  vH  A
)  vH  ( C  i^i  D ) ) ) ) )
1810, 17syl 15 . . . . . . . . 9  |-  ( x  e.  CH  ->  ( A. y  e.  CH  (
y  C_  D  ->  ( ( y  vH  C
)  i^i  D )  C_  ( y  vH  ( C  i^i  D ) ) )  ->  ( (
x  vH  A )  C_  D  ->  ( (
( x  vH  A
)  vH  C )  i^i  D )  C_  (
( x  vH  A
)  vH  ( C  i^i  D ) ) ) ) )
1918adantr 451 . . . . . . . 8  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( A. y  e.  CH  ( y 
C_  D  ->  (
( y  vH  C
)  i^i  D )  C_  ( y  vH  ( C  i^i  D ) ) )  ->  ( (
x  vH  A )  C_  D  ->  ( (
( x  vH  A
)  vH  C )  i^i  D )  C_  (
( x  vH  A
)  vH  ( C  i^i  D ) ) ) ) )
204, 5, 2, 3mdslmd1lem3 22962 . . . . . . . 8  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( (
( x  vH  A
)  C_  D  ->  ( ( ( x  vH  A )  vH  C
)  i^i  D )  C_  ( ( x  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  x  /\  x  C_  ( D  i^i  B
) )  ->  (
( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )
2119, 20syld 40 . . . . . . 7  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( A. y  e.  CH  ( y 
C_  D  ->  (
( y  vH  C
)  i^i  D )  C_  ( y  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  x  /\  x  C_  ( D  i^i  B
) )  ->  (
( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )
2221ex 423 . . . . . 6  |-  ( x  e.  CH  ->  (
( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  D  ->  ( ( y  vH  C )  i^i 
D )  C_  (
y  vH  ( C  i^i  D ) ) )  ->  ( ( ( ( C  i^i  B
)  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) )  ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) ) )
2322com3l 75 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  D  ->  ( ( y  vH  C )  i^i 
D )  C_  (
y  vH  ( C  i^i  D ) ) )  ->  ( x  e. 
CH  ->  ( ( ( ( C  i^i  B
)  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) )  ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) ) )
2423ralrimdv 2666 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  D  ->  ( ( y  vH  C )  i^i 
D )  C_  (
y  vH  ( C  i^i  D ) ) )  ->  A. x  e.  CH  ( ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) )  ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
25 mdbr2 22931 . . . . 5  |-  ( ( C  e.  CH  /\  D  e.  CH )  ->  ( C  MH  D  <->  A. y  e.  CH  (
y  C_  D  ->  ( ( y  vH  C
)  i^i  D )  C_  ( y  vH  ( C  i^i  D ) ) ) ) )
262, 3, 25mp2an 653 . . . 4  |-  ( C  MH  D  <->  A. y  e.  CH  ( y  C_  D  ->  ( ( y  vH  C )  i^i 
D )  C_  (
y  vH  ( C  i^i  D ) ) ) )
272, 5chincli 22094 . . . . 5  |-  ( C  i^i  B )  e. 
CH
283, 5chincli 22094 . . . . 5  |-  ( D  i^i  B )  e. 
CH
2927, 28mdsl2i 22957 . . . 4  |-  ( ( C  i^i  B )  MH  ( D  i^i  B )  <->  A. x  e.  CH  ( ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  x  /\  x  C_  ( D  i^i  B ) )  ->  ( ( x  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( x  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) )
3024, 26, 293imtr4g 261 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( C  MH  D  ->  ( C  i^i  B )  MH  ( D  i^i  B ) ) )
31 chincl 22133 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
325, 31mpan2 652 . . . . . . . . . 10  |-  ( x  e.  CH  ->  (
x  i^i  B )  e.  CH )
33 sseq1 3233 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  ( D  i^i  B )  <->  ( x  i^i  B )  C_  ( D  i^i  B ) ) )
34 oveq1 5907 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( C  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( C  i^i  B ) ) )
3534ineq1d 3403 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  =  ( ( ( x  i^i 
B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) )
36 oveq1 5907 . . . . . . . . . . . . 13  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) )  =  ( ( x  i^i 
B )  vH  (
( C  i^i  B
)  i^i  ( D  i^i  B ) ) ) )
3735, 36sseq12d 3241 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) )  <->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) )
3833, 37imbi12d 311 . . . . . . . . . . 11  |-  ( y  =  ( x  i^i 
B )  ->  (
( y  C_  ( D  i^i  B )  -> 
( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  <->  ( (
x  i^i  B )  C_  ( D  i^i  B
)  ->  ( (
( x  i^i  B
)  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
( x  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )
3938rspcv 2914 . . . . . . . . . 10  |-  ( ( x  i^i  B )  e.  CH  ->  ( A. y  e.  CH  (
y  C_  ( D  i^i  B )  ->  (
( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( x  i^i  B )  C_  ( D  i^i  B )  ->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
4032, 39syl 15 . . . . . . . . 9  |-  ( x  e.  CH  ->  ( A. y  e.  CH  (
y  C_  ( D  i^i  B )  ->  (
( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( x  i^i  B )  C_  ( D  i^i  B )  ->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
4140adantr 451 . . . . . . . 8  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( A. y  e.  CH  ( y 
C_  ( D  i^i  B )  ->  ( (
y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( x  i^i  B )  C_  ( D  i^i  B )  ->  ( ( ( x  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( x  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
424, 5, 2, 3mdslmd1lem4 22963 . . . . . . . 8  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( (
( x  i^i  B
)  C_  ( D  i^i  B )  ->  (
( ( x  i^i 
B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
( x  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( ( C  i^i  D ) 
C_  x  /\  x  C_  D )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) ) )
4341, 42syld 40 . . . . . . 7  |-  ( ( x  e.  CH  /\  ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) ) )  ->  ( A. y  e.  CH  ( y 
C_  ( D  i^i  B )  ->  ( (
y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( ( C  i^i  D ) 
C_  x  /\  x  C_  D )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) ) )
4443ex 423 . . . . . 6  |-  ( x  e.  CH  ->  (
( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  ( D  i^i  B )  ->  ( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
( ( C  i^i  D )  C_  x  /\  x  C_  D )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) ) )
4544com3l 75 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  ( D  i^i  B )  ->  ( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
x  e.  CH  ->  ( ( ( C  i^i  D )  C_  x  /\  x  C_  D )  -> 
( ( x  vH  C )  i^i  D
)  C_  ( x  vH  ( C  i^i  D
) ) ) ) ) )
4645ralrimdv 2666 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( A. y  e.  CH  ( y  C_  ( D  i^i  B )  ->  ( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  A. x  e.  CH  ( ( ( C  i^i  D ) 
C_  x  /\  x  C_  D )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) ) )
47 mdbr2 22931 . . . . 5  |-  ( ( ( C  i^i  B
)  e.  CH  /\  ( D  i^i  B )  e.  CH )  -> 
( ( C  i^i  B )  MH  ( D  i^i  B )  <->  A. y  e.  CH  ( y  C_  ( D  i^i  B )  ->  ( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
4827, 28, 47mp2an 653 . . . 4  |-  ( ( C  i^i  B )  MH  ( D  i^i  B )  <->  A. y  e.  CH  ( y  C_  ( D  i^i  B )  -> 
( ( y  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( y  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) )
492, 3mdsl2i 22957 . . . 4  |-  ( C  MH  D  <->  A. x  e.  CH  ( ( ( C  i^i  D ) 
C_  x  /\  x  C_  D )  ->  (
( x  vH  C
)  i^i  D )  C_  ( x  vH  ( C  i^i  D ) ) ) )
5046, 48, 493imtr4g 261 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( C  i^i  B )  MH  ( D  i^i  B
)  ->  C  MH  D ) )
5130, 50impbid 183 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( C  MH  D 
<->  ( C  i^i  B
)  MH  ( D  i^i  B ) ) )
528, 51sylan2br 462 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( C  i^i  D )  /\  ( C  vH  D )  C_  ( A  vH  B ) ) )  ->  ( C  MH  D  <->  ( C  i^i  B )  MH  ( D  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    i^i cin 3185    C_ wss 3186   class class class wbr 4060  (class class class)co 5900   CHcch 21564    vH chj 21568    MH cmd 21601    MH* cdmd 21602
This theorem is referenced by:  mdslmd2i  22965  mdcompli  23064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cc 8106  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862  ax-hilex 21634  ax-hfvadd 21635  ax-hvcom 21636  ax-hvass 21637  ax-hv0cl 21638  ax-hvaddid 21639  ax-hfvmul 21640  ax-hvmulid 21641  ax-hvmulass 21642  ax-hvdistr1 21643  ax-hvdistr2 21644  ax-hvmul0 21645  ax-hfi 21713  ax-his1 21716  ax-his2 21717  ax-his3 21718  ax-his4 21719  ax-hcompl 21836
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-omul 6526  df-er 6702  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-acn 7620  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-clim 12009  df-rlim 12010  df-sum 12206  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-mulg 14541  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-cn 17013  df-cnp 17014  df-lm 17015  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cfil 18734  df-cau 18735  df-cmet 18736  df-grpo 20911  df-gid 20912  df-ginv 20913  df-gdiv 20914  df-ablo 21002  df-subgo 21022  df-vc 21157  df-nv 21203  df-va 21206  df-ba 21207  df-sm 21208  df-0v 21209  df-vs 21210  df-nmcv 21211  df-ims 21212  df-dip 21329  df-ssp 21353  df-ph 21446  df-cbn 21497  df-hnorm 21603  df-hba 21604  df-hvsub 21606  df-hlim 21607  df-hcau 21608  df-sh 21841  df-ch 21856  df-oc 21886  df-ch0 21887  df-shs 21942  df-chj 21944  df-md 22915  df-dmd 22916
  Copyright terms: Public domain W3C validator