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Theorem mdslmd1i 27808
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 3690 . . 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 26936 . . . 4  |-  ( A  vH  B )  e. 
CH
72, 3, 6chlubi 26950 . . 3  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  <->  ( C  vH  D )  C_  ( A  vH  B ) )
81, 7anbi12i 701 . 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 26836 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  A  e.  CH )  ->  ( x  vH  A
)  e.  CH )
104, 9mpan2 675 . . . . . . . . . 10  |-  ( x  e.  CH  ->  (
x  vH  A )  e.  CH )
11 sseq1 3491 . . . . . . . . . . . 12  |-  ( y  =  ( x  vH  A )  ->  (
y  C_  D  <->  ( x  vH  A )  C_  D
) )
12 oveq1 6312 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  C )  =  ( ( x  vH  A )  vH  C ) )
1312ineq1d 3669 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
( y  vH  C
)  i^i  D )  =  ( ( ( x  vH  A )  vH  C )  i^i 
D ) )
14 oveq1 6312 . . . . . . . . . . . . 13  |-  ( y  =  ( x  vH  A )  ->  (
y  vH  ( C  i^i  D ) )  =  ( ( x  vH  A )  vH  ( C  i^i  D ) ) )
1513, 14sseq12d 3499 . . . . . . . . . . . 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 321 . . . . . . . . . . 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 3184 . . . . . . . . . 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 17 . . . . . . . . 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 466 . . . . . . . 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 27806 . . . . . . . 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 45 . . . . . . 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 435 . . . . . 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 84 . . . . 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 2848 . . . 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 27775 . . . . 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 676 . . . 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 26939 . . . . 5  |-  ( C  i^i  B )  e. 
CH
283, 5chincli 26939 . . . . 5  |-  ( D  i^i  B )  e. 
CH
2927, 28mdsl2i 27801 . . . 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 273 . . 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 26978 . . . . . . . . . . 11  |-  ( ( x  e.  CH  /\  B  e.  CH )  ->  ( x  i^i  B
)  e.  CH )
325, 31mpan2 675 . . . . . . . . . 10  |-  ( x  e.  CH  ->  (
x  i^i  B )  e.  CH )
33 sseq1 3491 . . . . . . . . . . . 12  |-  ( y  =  ( x  i^i 
B )  ->  (
y  C_  ( D  i^i  B )  <->  ( x  i^i  B )  C_  ( D  i^i  B ) ) )
34 oveq1 6312 . . . . . . . . . . . . . 14  |-  ( y  =  ( x  i^i 
B )  ->  (
y  vH  ( C  i^i  B ) )  =  ( ( x  i^i 
B )  vH  ( C  i^i  B ) ) )
3534ineq1d 3669 . . . . . . . . . . . . 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 6312 . . . . . . . . . . . . 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 3499 . . . . . . . . . . . 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 321 . . . . . . . . . . 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 3184 . . . . . . . . . 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 17 . . . . . . . . 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 466 . . . . . . . 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 27807 . . . . . . . 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 45 . . . . . . 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 435 . . . . . 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 84 . . . . 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 2848 . . . 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 27775 . . . . 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 676 . . . 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 27801 . . . 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 273 . . 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 193 . 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 478 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    i^i cin 3441    C_ wss 3442   class class class wbr 4426  (class class class)co 6305   CHcch 26408    vH chj 26412    MH cmd 26445    MH* cdmd 26446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cc 8863  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618  ax-hilex 26478  ax-hfvadd 26479  ax-hvcom 26480  ax-hvass 26481  ax-hv0cl 26482  ax-hvaddid 26483  ax-hfvmul 26484  ax-hvmulid 26485  ax-hvmulass 26486  ax-hvdistr1 26487  ax-hvdistr2 26488  ax-hvmul0 26489  ax-hfi 26558  ax-his1 26561  ax-his2 26562  ax-his3 26563  ax-his4 26564  ax-hcompl 26681
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-omul 7195  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-acn 8375  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-struct 15077  df-ndx 15078  df-slot 15079  df-base 15080  df-sets 15081  df-ress 15082  df-plusg 15156  df-mulr 15157  df-starv 15158  df-sca 15159  df-vsca 15160  df-ip 15161  df-tset 15162  df-ple 15163  df-ds 15165  df-unif 15166  df-hom 15167  df-cco 15168  df-rest 15271  df-topn 15272  df-0g 15290  df-gsum 15291  df-topgen 15292  df-pt 15293  df-prds 15296  df-xrs 15350  df-qtop 15355  df-imas 15356  df-xps 15358  df-mre 15434  df-mrc 15435  df-acs 15437  df-mgm 16430  df-sgrp 16469  df-mnd 16479  df-submnd 16525  df-mulg 16618  df-cntz 16913  df-cmn 17358  df-psmet 18888  df-xmet 18889  df-met 18890  df-bl 18891  df-mopn 18892  df-fbas 18893  df-fg 18894  df-cnfld 18897  df-top 19843  df-bases 19844  df-topon 19845  df-topsp 19846  df-cld 19956  df-ntr 19957  df-cls 19958  df-nei 20036  df-cn 20165  df-cnp 20166  df-lm 20167  df-haus 20253  df-tx 20499  df-hmeo 20692  df-fil 20783  df-fm 20875  df-flim 20876  df-flf 20877  df-xms 21257  df-ms 21258  df-tms 21259  df-cfil 22109  df-cau 22110  df-cmet 22111  df-grpo 25755  df-gid 25756  df-ginv 25757  df-gdiv 25758  df-ablo 25846  df-subgo 25866  df-vc 26001  df-nv 26047  df-va 26050  df-ba 26051  df-sm 26052  df-0v 26053  df-vs 26054  df-nmcv 26055  df-ims 26056  df-dip 26173  df-ssp 26197  df-ph 26290  df-cbn 26341  df-hnorm 26447  df-hba 26448  df-hvsub 26450  df-hlim 26451  df-hcau 26452  df-sh 26686  df-ch 26700  df-oc 26731  df-ch0 26732  df-shs 26787  df-chj 26789  df-md 27759  df-dmd 27760
This theorem is referenced by:  mdslmd2i  27809  mdcompli  27908
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