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Theorem mdslmd1i 22905
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 3392 . . 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 22032 . . . 4  |-  ( A  vH  B )  e. 
CH
72, 3, 6chlubi 22046 . . 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 21932 . . . . . . . . . . 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 3200 . . . . . . . . . . . 12  |-  ( y  =  ( x  vH  A )  ->  (
y  C_  D  <->  ( x  vH  A )  C_  D
) )
12 oveq1 5827 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  C )  =  ( ( x  vH  A )  vH  C ) )
1312ineq1d 3370 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
( y  vH  C
)  i^i  D )  =  ( ( ( x  vH  A )  vH  C )  i^i 
D ) )
14 oveq1 5827 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  ( C  i^i  D ) )  =  ( ( x  vH  A )  vH  ( C  i^i  D ) ) )
1513, 14sseq12d 3208 . . . . . . . . . . . 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 2881 . . . . . . . . . 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 22903 . . . . . . . 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 2633 . . . 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 22872 . . . . 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 22035 . . . . 5  |-  ( C  i^i  B )  e. 
CH
283, 5chincli 22035 . . . . 5  |-  ( D  i^i  B )  e. 
CH
2927, 28mdsl2i 22898 . . . 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 22074 . . . . . . . . . . 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 3200 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  ( D  i^i  B )  <->  ( x  i^i  B )  C_  ( D  i^i  B ) ) )
34 oveq1 5827 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( C  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( C  i^i  B ) ) )
3534ineq1d 3370 . . . . . . . . . . . . 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 5827 . . . . . . . . . . . . 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 3208 . . . . . . . . . . . 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 2881 . . . . . . . . . 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 22904 . . . . . . . 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 2633 . . . 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 22872 . . . . 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 22898 . . . 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 1623    e. wcel 1685   A.wral 2544    i^i cin 3152    C_ wss 3153   class class class wbr 4024  (class class class)co 5820   CHcch 21505    vH chj 21509    MH cmd 21542    MH* cdmd 21543
This theorem is referenced by:  mdslmd2i  22906  mdcompli  23005
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-inf2 7338  ax-cc 8057  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  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813  ax-hilex 21575  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvmulass 21583  ax-hvdistr1 21584  ax-hvdistr2 21585  ax-hvmul0 21586  ax-hfi 21654  ax-his1 21657  ax-his2 21658  ax-his3 21659  ax-his4 21660  ax-hcompl 21777
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 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-se 4352  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-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-omul 6480  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-acn 7571  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-seq 11043  df-exp 11101  df-hash 11334  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-clim 11958  df-rlim 11959  df-sum 12155  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-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-cn 16953  df-cnp 16954  df-lm 16955  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cfil 18677  df-cau 18678  df-cmet 18679  df-grpo 20852  df-gid 20853  df-ginv 20854  df-gdiv 20855  df-ablo 20943  df-subgo 20963  df-vc 21096  df-nv 21142  df-va 21145  df-ba 21146  df-sm 21147  df-0v 21148  df-vs 21149  df-nmcv 21150  df-ims 21151  df-dip 21268  df-ssp 21292  df-ph 21385  df-cbn 21436  df-hnorm 21544  df-hba 21545  df-hvsub 21547  df-hlim 21548  df-hcau 21549  df-sh 21782  df-ch 21797  df-oc 21827  df-ch0 21828  df-shs 21883  df-chj 21885  df-md 22856  df-dmd 22857
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