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Theorem mdsl1i 28022
Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdsl.1  |-  A  e. 
CH
mdsl.2  |-  B  e. 
CH
Assertion
Ref Expression
mdsl1i  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  <-> 
A  MH  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem mdsl1i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sseq2 3465 . . . . . . . 8  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( ( A  i^i  B )  C_  x 
<->  ( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) ) ) )
2 sseq1 3464 . . . . . . . 8  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( x  C_  ( A  vH  B
)  <->  ( y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )
31, 2anbi12d 722 . . . . . . 7  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( (
( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  <->  ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B ) )  /\  ( y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) ) )
4 sseq1 3464 . . . . . . . 8  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( x  C_  B  <->  ( y  vH  ( A  i^i  B ) )  C_  B )
)
5 oveq1 6321 . . . . . . . . . 10  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( x  vH  A )  =  ( ( y  vH  ( A  i^i  B ) )  vH  A ) )
65ineq1d 3644 . . . . . . . . 9  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B ) )
7 oveq1 6321 . . . . . . . . 9  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( x  vH  ( A  i^i  B
) )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) )
86, 7eqeq12d 2476 . . . . . . . 8  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( (
( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) )  <->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) )
94, 8imbi12d 326 . . . . . . 7  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  <->  ( (
y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) )
103, 9imbi12d 326 . . . . . 6  |-  ( x  =  ( y  vH  ( A  i^i  B ) )  ->  ( (
( ( A  i^i  B )  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  <-> 
( ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B ) )  /\  ( y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )  -> 
( ( y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) ) )
1110rspccv 3158 . . . . 5  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  ->  ( ( y  vH  ( A  i^i  B ) )  e.  CH  ->  ( ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B ) )  /\  ( y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )  -> 
( ( y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) ) )
12 impexp 452 . . . . . . 7  |-  ( ( ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  /\  (
y  vH  ( A  i^i  B ) )  C_  B )  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) )  <->  ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  (
( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  ->  (
( y  vH  ( A  i^i  B ) ) 
C_  B  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) )
13 impexp 452 . . . . . . 7  |-  ( ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  ->  (
( y  vH  ( A  i^i  B ) ) 
C_  B  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) )  <->  ( (
y  vH  ( A  i^i  B ) )  e. 
CH  ->  ( ( ( A  i^i  B ) 
C_  ( y  vH  ( A  i^i  B ) )  /\  ( y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )  ->  ( ( y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A
)  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) ) )
1412, 13bitr2i 258 . . . . . 6  |-  ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  ->  (
( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )  ->  ( (
y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) )  <->  ( ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  (
( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  /\  (
y  vH  ( A  i^i  B ) )  C_  B )  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) )
15 inss2 3664 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  B
16 mdsl.1 . . . . . . . . . . . . . . 15  |-  A  e. 
CH
17 mdsl.2 . . . . . . . . . . . . . . 15  |-  B  e. 
CH
1816, 17chincli 27161 . . . . . . . . . . . . . 14  |-  ( A  i^i  B )  e. 
CH
19 chlub 27210 . . . . . . . . . . . . . 14  |-  ( ( y  e.  CH  /\  ( A  i^i  B )  e.  CH  /\  B  e.  CH )  ->  (
( y  C_  B  /\  ( A  i^i  B
)  C_  B )  <->  ( y  vH  ( A  i^i  B ) ) 
C_  B ) )
2018, 17, 19mp3an23 1365 . . . . . . . . . . . . 13  |-  ( y  e.  CH  ->  (
( y  C_  B  /\  ( A  i^i  B
)  C_  B )  <->  ( y  vH  ( A  i^i  B ) ) 
C_  B ) )
2120biimpd 212 . . . . . . . . . . . 12  |-  ( y  e.  CH  ->  (
( y  C_  B  /\  ( A  i^i  B
)  C_  B )  ->  ( y  vH  ( A  i^i  B ) ) 
C_  B ) )
2215, 21mpan2i 688 . . . . . . . . . . 11  |-  ( y  e.  CH  ->  (
y  C_  B  ->  ( y  vH  ( A  i^i  B ) ) 
C_  B ) )
2317, 16chub2i 27171 . . . . . . . . . . . 12  |-  B  C_  ( A  vH  B )
24 sstr 3451 . . . . . . . . . . . 12  |-  ( ( ( y  vH  ( A  i^i  B ) ) 
C_  B  /\  B  C_  ( A  vH  B
) )  ->  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )
2523, 24mpan2 682 . . . . . . . . . . 11  |-  ( ( y  vH  ( A  i^i  B ) ) 
C_  B  ->  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )
2622, 25syl6 34 . . . . . . . . . 10  |-  ( y  e.  CH  ->  (
y  C_  B  ->  ( y  vH  ( A  i^i  B ) ) 
C_  ( A  vH  B ) ) )
27 chub2 27209 . . . . . . . . . . 11  |-  ( ( ( A  i^i  B
)  e.  CH  /\  y  e.  CH )  ->  ( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) ) )
2818, 27mpan 681 . . . . . . . . . 10  |-  ( y  e.  CH  ->  ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B ) ) )
2926, 28jctild 550 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
y  C_  B  ->  ( ( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) ) )
30 chjcl 27058 . . . . . . . . . 10  |-  ( ( y  e.  CH  /\  ( A  i^i  B )  e.  CH )  -> 
( y  vH  ( A  i^i  B ) )  e.  CH )
3118, 30mpan2 682 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
y  vH  ( A  i^i  B ) )  e. 
CH )
3229, 31jctild 550 . . . . . . . 8  |-  ( y  e.  CH  ->  (
y  C_  B  ->  ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  (
( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) ) ) )
3332, 22jcad 540 . . . . . . 7  |-  ( y  e.  CH  ->  (
y  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  /\  (
y  vH  ( A  i^i  B ) )  C_  B ) ) )
34 chjass 27234 . . . . . . . . . . . 12  |-  ( ( y  e.  CH  /\  ( A  i^i  B )  e.  CH  /\  A  e.  CH )  ->  (
( y  vH  ( A  i^i  B ) )  vH  A )  =  ( y  vH  (
( A  i^i  B
)  vH  A )
) )
3518, 16, 34mp3an23 1365 . . . . . . . . . . 11  |-  ( y  e.  CH  ->  (
( y  vH  ( A  i^i  B ) )  vH  A )  =  ( y  vH  (
( A  i^i  B
)  vH  A )
) )
3618, 16chjcomi 27169 . . . . . . . . . . . . 13  |-  ( ( A  i^i  B )  vH  A )  =  ( A  vH  ( A  i^i  B ) )
3716, 17chabs1i 27219 . . . . . . . . . . . . 13  |-  ( A  vH  ( A  i^i  B ) )  =  A
3836, 37eqtri 2483 . . . . . . . . . . . 12  |-  ( ( A  i^i  B )  vH  A )  =  A
3938oveq2i 6325 . . . . . . . . . . 11  |-  ( y  vH  ( ( A  i^i  B )  vH  A ) )  =  ( y  vH  A
)
4035, 39syl6eq 2511 . . . . . . . . . 10  |-  ( y  e.  CH  ->  (
( y  vH  ( A  i^i  B ) )  vH  A )  =  ( y  vH  A
) )
4140ineq1d 3644 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  A )  i^i  B
) )
42 chjass 27234 . . . . . . . . . . 11  |-  ( ( y  e.  CH  /\  ( A  i^i  B )  e.  CH  /\  ( A  i^i  B )  e. 
CH )  ->  (
( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) )  =  ( y  vH  ( ( A  i^i  B )  vH  ( A  i^i  B ) ) ) )
4318, 18, 42mp3an23 1365 . . . . . . . . . 10  |-  ( y  e.  CH  ->  (
( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) )  =  ( y  vH  ( ( A  i^i  B )  vH  ( A  i^i  B ) ) ) )
4418chjidmi 27222 . . . . . . . . . . 11  |-  ( ( A  i^i  B )  vH  ( A  i^i  B ) )  =  ( A  i^i  B )
4544oveq2i 6325 . . . . . . . . . 10  |-  ( y  vH  ( ( A  i^i  B )  vH  ( A  i^i  B ) ) )  =  ( y  vH  ( A  i^i  B ) )
4643, 45syl6eq 2511 . . . . . . . . 9  |-  ( y  e.  CH  ->  (
( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) )  =  ( y  vH  ( A  i^i  B ) ) )
4741, 46eqeq12d 2476 . . . . . . . 8  |-  ( y  e.  CH  ->  (
( ( ( y  vH  ( A  i^i  B ) )  vH  A
)  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) )  <-> 
( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) ) ) )
4847biimpd 212 . . . . . . 7  |-  ( y  e.  CH  ->  (
( ( ( y  vH  ( A  i^i  B ) )  vH  A
)  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) )  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) ) )
4933, 48imim12d 77 . . . . . 6  |-  ( y  e.  CH  ->  (
( ( ( ( y  vH  ( A  i^i  B ) )  e.  CH  /\  (
( A  i^i  B
)  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) ) )  /\  (
y  vH  ( A  i^i  B ) )  C_  B )  ->  (
( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i  B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) )  ->  ( y  C_  B  ->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
5014, 49syl5bi 225 . . . . 5  |-  ( y  e.  CH  ->  (
( ( y  vH  ( A  i^i  B ) )  e.  CH  ->  ( ( ( A  i^i  B )  C_  ( y  vH  ( A  i^i  B
) )  /\  (
y  vH  ( A  i^i  B ) )  C_  ( A  vH  B ) )  ->  ( (
y  vH  ( A  i^i  B ) )  C_  B  ->  ( ( ( y  vH  ( A  i^i  B ) )  vH  A )  i^i 
B )  =  ( ( y  vH  ( A  i^i  B ) )  vH  ( A  i^i  B ) ) ) ) )  ->  ( y  C_  B  ->  ( (
y  vH  A )  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
5111, 50syl5com 31 . . . 4  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  ->  ( y  e. 
CH  ->  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
5251ralrimiv 2811 . . 3  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  ->  A. y  e.  CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i  B
)  =  ( y  vH  ( A  i^i  B ) ) ) )
53 mdbr 27995 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. y  e.  CH  (
y  C_  B  ->  ( ( y  vH  A
)  i^i  B )  =  ( y  vH  ( A  i^i  B ) ) ) ) )
5416, 17, 53mp2an 683 . . 3  |-  ( A  MH  B  <->  A. y  e.  CH  ( y  C_  B  ->  ( ( y  vH  A )  i^i 
B )  =  ( y  vH  ( A  i^i  B ) ) ) )
5552, 54sylibr 217 . 2  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  ->  A  MH  B
)
56 mdbr 27995 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
5716, 17, 56mp2an 683 . . 3  |-  ( A  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  A )  i^i 
B )  =  ( x  vH  ( A  i^i  B ) ) ) )
58 ax-1 6 . . . 4  |-  ( ( x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  ->  (
( ( A  i^i  B )  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
5958ralimi 2792 . . 3  |-  ( A. x  e.  CH  ( x 
C_  B  ->  (
( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )  ->  A. x  e.  CH  ( ( ( A  i^i  B ) 
C_  x  /\  x  C_  ( A  vH  B
) )  ->  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6057, 59sylbi 200 . 2  |-  ( A  MH  B  ->  A. x  e.  CH  ( ( ( A  i^i  B ) 
C_  x  /\  x  C_  ( A  vH  B
) )  ->  (
x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6155, 60impbii 192 1  |-  ( A. x  e.  CH  ( ( ( A  i^i  B
)  C_  x  /\  x  C_  ( A  vH  B ) )  -> 
( x  C_  B  ->  ( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) )  <-> 
A  MH  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748    i^i cin 3414    C_ wss 3415   class class class wbr 4415  (class class class)co 6314   CHcch 26630    vH chj 26634    MH cmd 26667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cc 8890  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644  ax-hilex 26700  ax-hfvadd 26701  ax-hvcom 26702  ax-hvass 26703  ax-hv0cl 26704  ax-hvaddid 26705  ax-hfvmul 26706  ax-hvmulid 26707  ax-hvmulass 26708  ax-hvdistr1 26709  ax-hvdistr2 26710  ax-hvmul0 26711  ax-hfi 26780  ax-his1 26783  ax-his2 26784  ax-his3 26785  ax-his4 26786  ax-hcompl 26903
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-omul 7212  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-fi 7950  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-acn 8401  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-q 11293  df-rp 11331  df-xneg 11437  df-xadd 11438  df-xmul 11439  df-ioo 11667  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-fl 12059  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-rlim 13601  df-sum 13801  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-rest 15369  df-topn 15370  df-0g 15388  df-gsum 15389  df-topgen 15390  df-pt 15391  df-prds 15394  df-xrs 15448  df-qtop 15454  df-imas 15455  df-xps 15458  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-mulg 16724  df-cntz 17019  df-cmn 17480  df-psmet 19010  df-xmet 19011  df-met 19012  df-bl 19013  df-mopn 19014  df-fbas 19015  df-fg 19016  df-cnfld 19019  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cld 20082  df-ntr 20083  df-cls 20084  df-nei 20162  df-cn 20291  df-cnp 20292  df-lm 20293  df-haus 20379  df-tx 20625  df-hmeo 20818  df-fil 20909  df-fm 21001  df-flim 21002  df-flf 21003  df-xms 21383  df-ms 21384  df-tms 21385  df-cfil 22273  df-cau 22274  df-cmet 22275  df-grpo 25967  df-gid 25968  df-ginv 25969  df-gdiv 25970  df-ablo 26058  df-subgo 26078  df-vc 26213  df-nv 26259  df-va 26262  df-ba 26263  df-sm 26264  df-0v 26265  df-vs 26266  df-nmcv 26267  df-ims 26268  df-dip 26385  df-ssp 26409  df-ph 26502  df-cbn 26553  df-hnorm 26669  df-hba 26670  df-hvsub 26672  df-hlim 26673  df-hcau 26674  df-sh 26908  df-ch 26922  df-oc 26953  df-ch0 26954  df-shs 27009  df-chj 27011  df-md 27981
This theorem is referenced by:  mdsl2i  28023  cvmdi  28025
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