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Theorem fh2 25169
Description: Foulis-Holland Theorem. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Second of two parts. Theorem 5 of [Kalmbach] p. 25. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
fh2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( A  i^i  B
)  vH  ( A  i^i  C ) ) )

Proof of Theorem fh2
StepHypRef Expression
1 chincl 25049 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
2 chincl 25049 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
3 chjcl 24907 . . . . . . . 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 477 . . . . . . 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 826 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
6 chjcl 24907 . . . . . . . 8  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
7 chincl 25049 . . . . . . . 8  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  i^i  ( B  vH  C ) )  e.  CH )
86, 7sylan2 474 . . . . . . 7  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  CH )
9 chsh 24774 . . . . . . 7  |-  ( ( A  i^i  ( B  vH  C ) )  e.  CH  ->  ( A  i^i  ( B  vH  C ) )  e.  SH )
108, 9syl 16 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  SH )
115, 10jca 532 . . . . 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 1184 . . . 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 465 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( (
( A  i^i  B
)  vH  ( A  i^i  C ) )  e. 
CH  /\  ( A  i^i  ( B  vH  C
) )  e.  SH ) )
14 ledi 25090 . . . 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 465 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  C_  ( A  i^i  ( B  vH  C
) ) )
16 chdmj1 25079 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) ) )
171, 2, 16syl2an 477 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )
18 chdmm1 25075 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
1918adantr 465 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( _|_ `  ( A  i^i  B
) )  =  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )
2019ineq1d 3654 . . . . . . . . . 10  |-  ( ( ( 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  i^i  C ) ) ) )
2117, 20eqtrd 2493 . . . . . . . . 9  |-  ( ( ( 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  i^i  C ) ) ) )
22213impdi 1274 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  ( _|_ `  ( ( A  i^i  B )  vH  ( A  i^i  C ) ) )  =  ( ( ( _|_ `  A
)  vH  ( _|_ `  B ) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
2322ineq2d 3655 . . . . . . 7  |-  ( ( 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 ) ) ) )  =  ( ( A  i^i  ( B  vH  C ) )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B
) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
2423adantr 465 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( A  i^i  ( B  vH  C ) )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B
) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
25 in4 3669 . . . . . . 7  |-  ( ( A  i^i  ( B  vH  C ) )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B
) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
26 cmcm2 25166 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  C_H  ( _|_ `  B
) ) )
27 cmcm 25164 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  C_H  A ) )
28 choccl 24856 . . . . . . . . . . . . . 14  |-  ( B  e.  CH  ->  ( _|_ `  B )  e. 
CH )
29 cmbr3 25158 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( _|_ `  B )  e.  CH )  -> 
( A  C_H  ( _|_ `  B )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3028, 29sylan2 474 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  ( _|_ `  B )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3126, 27, 303bitr3d 283 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( B  C_H  A  <->  ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B
) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3231biimpa 484 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) )
33 incom 3646 . . . . . . . . . . 11  |-  ( A  i^i  ( _|_ `  B
) )  =  ( ( _|_ `  B
)  i^i  A )
3432, 33syl6eq 2509 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  B )  i^i  A
) )
35343adantl3 1146 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  B )  i^i  A
) )
3635adantrr 716 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  B )  i^i  A
) )
3736ineq1d 3654 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B
)  i^i  A )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
3825, 37syl5eq 2505 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( ( ( _|_ `  A )  vH  ( _|_ `  B ) )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B
)  i^i  A )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
3924, 38eqtrd 2493 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B
)  i^i  A )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
40 in4 3669 . . . . 5  |-  ( ( ( _|_ `  B
)  i^i  A )  i^i  ( ( B  vH  C )  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B )  i^i  ( B  vH  C ) )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
4139, 40syl6eq 2509 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B
)  i^i  ( B  vH  C ) )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
42 ococ 24956 . . . . . . . . . . 11  |-  ( B  e.  CH  ->  ( _|_ `  ( _|_ `  B
) )  =  B )
4342oveq1d 6210 . . . . . . . . . 10  |-  ( B  e.  CH  ->  (
( _|_ `  ( _|_ `  B ) )  vH  C )  =  ( B  vH  C
) )
4443ineq2d 3655 . . . . . . . . 9  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  ( B  vH  C ) ) )
45443ad2ant2 1010 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  ( B  vH  C ) ) )
4645adantr 465 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( _|_ `  B )  i^i  ( ( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  ( B  vH  C ) ) )
47 cmcm3 25165 . . . . . . . . . . 11  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  C_H  C  <->  ( _|_ `  B )  C_H  C ) )
48 cmbr3 25158 . . . . . . . . . . . 12  |-  ( ( ( _|_ `  B
)  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  B
)  C_H  C  <->  ( ( _|_ `  B )  i^i  ( ( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) ) )
4928, 48sylan 471 . . . . . . . . . . 11  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  B
)  C_H  C  <->  ( ( _|_ `  B )  i^i  ( ( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) ) )
5047, 49bitrd 253 . . . . . . . . . 10  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  C_H  C  <->  ( ( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  C )
) )
5150biimpa 484 . . . . . . . . 9  |-  ( ( ( B  e.  CH  /\  C  e.  CH )  /\  B  C_H  C )  ->  ( ( _|_ `  B )  i^i  (
( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) )
52513adantl1 1144 . . . . . . . 8  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  B  C_H  C )  ->  ( ( _|_ `  B )  i^i  (
( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) )
5352adantrl 715 . . . . . . 7  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( _|_ `  B )  i^i  ( ( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) )
5446, 53eqtr3d 2495 . . . . . 6  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( _|_ `  B )  i^i  ( B  vH  C
) )  =  ( ( _|_ `  B
)  i^i  C )
)
5554ineq1d 3654 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( (
( _|_ `  B
)  i^i  ( B  vH  C ) )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( ( _|_ `  B )  i^i  C )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
56 inass 3663 . . . . . . . . 9  |-  ( ( ( _|_ `  B
)  i^i  C )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( _|_ `  B )  i^i  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )
57 in12 3664 . . . . . . . . . . . 12  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
58 inass 3663 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  i^i  ( _|_ `  ( A  i^i  C ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
5957, 58eqtr4i 2484 . . . . . . . . . . 11  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( A  i^i  C
)  i^i  ( _|_ `  ( A  i^i  C
) ) )
60 chocin 25045 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  e.  CH  ->  (
( A  i^i  C
)  i^i  ( _|_ `  ( A  i^i  C
) ) )  =  0H )
612, 60syl 16 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  C )  i^i  ( _|_ `  ( A  i^i  C
) ) )  =  0H )
6259, 61syl5eq 2505 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
6362ineq2d 3655 . . . . . . . . 9  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  B
)  i^i  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) ) )  =  ( ( _|_ `  B )  i^i  0H ) )
6456, 63syl5eq 2505 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( ( _|_ `  B )  i^i  C
)  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( _|_ `  B
)  i^i  0H )
)
65643adant2 1007 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( ( _|_ `  B
)  i^i  C )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( _|_ `  B )  i^i  0H ) )
66 chm0 25041 . . . . . . . . 9  |-  ( ( _|_ `  B )  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6728, 66syl 16 . . . . . . . 8  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
68673ad2ant2 1010 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6965, 68eqtrd 2493 . . . . . 6  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( ( _|_ `  B
)  i^i  C )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
7069adantr 465 . . . . 5  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( (
( _|_ `  B
)  i^i  C )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
7155, 70eqtrd 2493 . . . 4  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( (
( _|_ `  B
)  i^i  ( B  vH  C ) )  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
7241, 71eqtrd 2493 . . 3  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  ( B  vH  C ) )  i^i  ( _|_ `  (
( A  i^i  B
)  vH  ( A  i^i  C ) ) ) )  =  0H )
73 pjoml 24986 . . 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 ) ) )
7413, 15, 72, 73syl12anc 1217 . 2  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  =  ( A  i^i  ( B  vH  C ) ) )
7574eqcomd 2460 1  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  /\  ( B  C_H  A  /\  B  C_H  C ) )  ->  ( A  i^i  ( B  vH  C
) )  =  ( ( A  i^i  B
)  vH  ( A  i^i  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    i^i cin 3430    C_ wss 3431   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   SHcsh 24477   CHcch 24478   _|_cort 24479    vH chj 24482   0Hc0h 24484    C_H ccm 24485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cc 8710  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468  ax-hilex 24548  ax-hfvadd 24549  ax-hvcom 24550  ax-hvass 24551  ax-hv0cl 24552  ax-hvaddid 24553  ax-hfvmul 24554  ax-hvmulid 24555  ax-hvmulass 24556  ax-hvdistr1 24557  ax-hvdistr2 24558  ax-hvmul0 24559  ax-hfi 24628  ax-his1 24631  ax-his2 24632  ax-his3 24633  ax-his4 24634  ax-hcompl 24751
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-omul 7030  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-acn 8218  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-clim 13079  df-rlim 13080  df-sum 13277  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-mulg 15662  df-cntz 15949  df-cmn 16395  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-cn 18958  df-cnp 18959  df-lm 18960  df-haus 19046  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-xms 20022  df-ms 20023  df-tms 20024  df-cfil 20893  df-cau 20894  df-cmet 20895  df-grpo 23825  df-gid 23826  df-ginv 23827  df-gdiv 23828  df-ablo 23916  df-subgo 23936  df-vc 24071  df-nv 24117  df-va 24120  df-ba 24121  df-sm 24122  df-0v 24123  df-vs 24124  df-nmcv 24125  df-ims 24126  df-dip 24243  df-ssp 24267  df-ph 24360  df-cbn 24411  df-hnorm 24517  df-hba 24518  df-hvsub 24520  df-hlim 24521  df-hcau 24522  df-sh 24756  df-ch 24771  df-oc 24802  df-ch0 24803  df-shs 24858  df-chj 24860  df-cm 25133
This theorem is referenced by:  fh2i  25172  atordi  25935  chirredlem2  25942
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