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Theorem fh1 22199
Description: Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. First of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
fh1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( A  i^i  B
)  vH  ( A  i^i  C ) ) )

Proof of Theorem fh1
StepHypRef Expression
1 chincl 22080 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
2 chincl 22080 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
3 chjcl 21938 . . . . . . . 8  |-  ( ( ( A  i^i  B
)  e.  CH  /\  ( A  i^i  C )  e.  CH )  -> 
( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
41, 2, 3syl2an 463 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
54anandis 803 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
6 chjcl 21938 . . . . . . . 8  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
7 chincl 22080 . . . . . . . 8  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  i^i  ( B  vH  C ) )  e.  CH )
86, 7sylan2 460 . . . . . . 7  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  CH )
9 chsh 21806 . . . . . . 7  |-  ( ( A  i^i  ( B  vH  C ) )  e.  CH  ->  ( A  i^i  ( B  vH  C ) )  e.  SH )
108, 9syl 15 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  SH )
115, 10jca 518 . . . . 5  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( (
( A  i^i  B
)  vH  ( A  i^i  C ) )  e. 
CH  /\  ( A  i^i  ( B  vH  C
) )  e.  SH ) )
12113impb 1147 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH  /\  ( A  i^i  ( B  vH  C ) )  e.  SH ) )
1312adantr 451 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( A  i^i  B
)  vH  ( A  i^i  C ) )  e. 
CH  /\  ( A  i^i  ( B  vH  C
) )  e.  SH ) )
14 ledi 22121 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  i^i  B
)  vH  ( A  i^i  C ) )  C_  ( A  i^i  ( B  vH  C ) ) )
1514adantr 451 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  C_  ( A  i^i  ( B  vH  C
) ) )
16 incom 3363 . . . . . . . 8  |-  ( A  i^i  ( B  vH  C ) )  =  ( ( B  vH  C )  i^i  A
)
1716a1i 10 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( B  vH  C
)  i^i  A )
)
18 chdmj1 22110 . . . . . . . . 9  |-  ( ( ( A  i^i  B
)  e.  CH  /\  ( A  i^i  C )  e.  CH )  -> 
( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) )  =  ( ( _|_ `  ( A  i^i  B
) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
191, 2, 18syl2an 463 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( _|_ `  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) )  =  ( ( _|_ `  ( A  i^i  B ) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
20 chdmm1 22106 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
21 chdmm1 22106 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( _|_ `  ( A  i^i  C ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) )
2220, 21ineqan12d 3374 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( ( _|_ `  ( A  i^i  B ) )  i^i  ( _|_ `  ( A  i^i  C ) ) )  =  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )
2319, 22eqtrd 2317 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( _|_ `  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) )  =  ( ( ( _|_ `  A
)  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) ) )
2417, 23ineq12d 3373 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( ( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) ) )
25243impdi 1237 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( ( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) ) )
2625adantr 451 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( ( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) ) )
27 inass 3381 . . . . . . 7  |-  ( ( ( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  ( ( B  vH  C
)  i^i  ( A  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) ) )
28 cmcm2 22197 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  C_H  ( _|_ `  B
) ) )
29 choccl 21887 . . . . . . . . . . . . . . 15  |-  ( B  e.  CH  ->  ( _|_ `  B )  e. 
CH )
30 cmbr3 22189 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CH  /\  ( _|_ `  B )  e.  CH )  -> 
( A  C_H  ( _|_ `  B )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3129, 30sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  ( _|_ `  B )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3228, 31bitrd 244 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B
) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3332biimpa 470 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  A  C_H  B )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) )
34333adantl3 1113 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_H  B )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) )
3534adantrr 697 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) )
36 cmcm2 22197 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  A  C_H  ( _|_ `  C
) ) )
37 choccl 21887 . . . . . . . . . . . . . . 15  |-  ( C  e.  CH  ->  ( _|_ `  C )  e. 
CH )
38 cmbr3 22189 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  CH  /\  ( _|_ `  C )  e.  CH )  -> 
( A  C_H  ( _|_ `  C )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) )  =  ( A  i^i  ( _|_ `  C ) ) ) )
3937, 38sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  ( _|_ `  C )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) )  =  ( A  i^i  ( _|_ `  C ) ) ) )
4036, 39bitrd 244 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  C_H  C  <->  ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C
) ) )  =  ( A  i^i  ( _|_ `  C ) ) ) )
4140biimpa 470 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) )  =  ( A  i^i  ( _|_ `  C ) ) )
42413adantl2 1112 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  A  C_H  C )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) )  =  ( A  i^i  ( _|_ `  C ) ) )
4342adantrl 696 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) )  =  ( A  i^i  ( _|_ `  C ) ) )
4435, 43ineq12d 3373 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )  i^i  ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  ( ( A  i^i  ( _|_ `  B ) )  i^i  ( A  i^i  ( _|_ `  C ) ) ) )
45 inindi 3388 . . . . . . . . 9  |-  ( A  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B
) )  i^i  (
( _|_ `  A
)  vH  ( _|_ `  C ) ) ) )  =  ( ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B
) ) )  i^i  ( A  i^i  (
( _|_ `  A
)  vH  ( _|_ `  C ) ) ) )
46 inindi 3388 . . . . . . . . 9  |-  ( A  i^i  ( ( _|_ `  B )  i^i  ( _|_ `  C ) ) )  =  ( ( A  i^i  ( _|_ `  B ) )  i^i  ( A  i^i  ( _|_ `  C ) ) )
4744, 45, 463eqtr4g 2342 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  ( A  i^i  ( ( _|_ `  B )  i^i  ( _|_ `  C
) ) ) )
4847ineq2d 3372 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( B  vH  C )  i^i  ( A  i^i  (
( ( _|_ `  A
)  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  C ) ) ) ) )  =  ( ( B  vH  C
)  i^i  ( A  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) ) )
4927, 48syl5eq 2329 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  ( ( B  vH  C
)  i^i  ( A  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) ) )
50 in12 3382 . . . . . 6  |-  ( ( B  vH  C )  i^i  ( A  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )  =  ( A  i^i  ( ( B  vH  C )  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )
5149, 50syl6eq 2333 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  ( A  i^i  ( ( B  vH  C )  i^i  ( ( _|_ `  B )  i^i  ( _|_ `  C ) ) ) ) )
52 chdmj1 22110 . . . . . . . . . . 11  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( _|_ `  ( B  vH  C ) )  =  ( ( _|_ `  B )  i^i  ( _|_ `  C ) ) )
5352ineq2d 3372 . . . . . . . . . 10  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( ( B  vH  C )  i^i  ( _|_ `  ( B  vH  C ) ) )  =  ( ( B  vH  C )  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )
54 chocin 22076 . . . . . . . . . . 11  |-  ( ( B  vH  C )  e.  CH  ->  (
( B  vH  C
)  i^i  ( _|_ `  ( B  vH  C
) ) )  =  0H )
556, 54syl 15 . . . . . . . . . 10  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( ( B  vH  C )  i^i  ( _|_ `  ( B  vH  C ) ) )  =  0H )
5653, 55eqtr3d 2319 . . . . . . . . 9  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( ( B  vH  C )  i^i  (
( _|_ `  B
)  i^i  ( _|_ `  C ) ) )  =  0H )
5756ineq2d 3372 . . . . . . . 8  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  (
( B  vH  C
)  i^i  ( ( _|_ `  B )  i^i  ( _|_ `  C
) ) ) )  =  ( A  i^i  0H ) )
58 chm0 22072 . . . . . . . 8  |-  ( A  e.  CH  ->  ( A  i^i  0H )  =  0H )
5957, 58sylan9eqr 2339 . . . . . . 7  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( ( B  vH  C )  i^i  (
( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )  =  0H )
60593impb 1147 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  ( ( B  vH  C )  i^i  ( ( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )  =  0H )
6160adantr 451 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( ( B  vH  C )  i^i  (
( _|_ `  B
)  i^i  ( _|_ `  C ) ) ) )  =  0H )
6251, 61eqtrd 2317 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( (
( B  vH  C
)  i^i  A )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( ( _|_ `  A )  vH  ( _|_ `  C ) ) ) )  =  0H )
6326, 62eqtrd 2317 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  0H )
64 pjoml 22017 . . 3  |-  ( ( ( ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH  /\  ( A  i^i  ( B  vH  C ) )  e.  SH )  /\  ( ( ( A  i^i  B )  vH  ( A  i^i  C ) )  C_  ( A  i^i  ( B  vH  C
) )  /\  (
( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  0H ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  =  ( A  i^i  ( B  vH  C ) ) )
6513, 15, 63, 64syl12anc 1180 . 2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  =  ( A  i^i  ( B  vH  C ) ) )
6665eqcomd 2290 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( A  C_H  B  /\  A  C_H  C ) )  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( A  i^i  B
)  vH  ( A  i^i  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    i^i cin 3153    C_ wss 3154   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   SHcsh 21510   CHcch 21511   _|_cort 21512    vH chj 21515   0Hc0h 21517    C_H ccm 21518
This theorem is referenced by:  cm2j  22201  fh1i  22202  chirredlem3  22974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cc 8063  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819  ax-hilex 21581  ax-hfvadd 21582  ax-hvcom 21583  ax-hvass 21584  ax-hv0cl 21585  ax-hvaddid 21586  ax-hfvmul 21587  ax-hvmulid 21588  ax-hvmulass 21589  ax-hvdistr1 21590  ax-hvdistr2 21591  ax-hvmul0 21592  ax-hfi 21660  ax-his1 21663  ax-his2 21664  ax-his3 21665  ax-his4 21666  ax-hcompl 21783
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-omul 6486  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-acn 7577  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-rlim 11965  df-sum 12161  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-cn 16959  df-cnp 16960  df-lm 16961  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cfil 18683  df-cau 18684  df-cmet 18685  df-grpo 20860  df-gid 20861  df-ginv 20862  df-gdiv 20863  df-ablo 20951  df-subgo 20971  df-vc 21104  df-nv 21150  df-va 21153  df-ba 21154  df-sm 21155  df-0v 21156  df-vs 21157  df-nmcv 21158  df-ims 21159  df-dip 21276  df-ssp 21300  df-ph 21393  df-cbn 21444  df-hnorm 21550  df-hba 21551  df-hvsub 21553  df-hlim 21554  df-hcau 21555  df-sh 21788  df-ch 21803  df-oc 21833  df-ch0 21834  df-shs 21889  df-chj 21891  df-cm 22164
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