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Theorem fh2 26951
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 26831 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
2 chincl 26831 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
3 chjcl 26689 . . . . . . . 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 475 . . . . . . 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 831 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
6 chjcl 26689 . . . . . . . 8  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
7 chincl 26831 . . . . . . . 8  |-  ( ( A  e.  CH  /\  ( B  vH  C )  e.  CH )  -> 
( A  i^i  ( B  vH  C ) )  e.  CH )
86, 7sylan2 472 . . . . . . 7  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  CH )
9 chsh 26556 . . . . . . 7  |-  ( ( A  i^i  ( B  vH  C ) )  e.  CH  ->  ( A  i^i  ( B  vH  C ) )  e.  SH )
108, 9syl 17 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( A  i^i  ( B  vH  C
) )  e.  SH )
115, 10jca 530 . . . . 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 1193 . . . 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 463 . . 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 26872 . . . 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 463 . . 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 26861 . . . . . . . . . . 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 475 . . . . . . . . . 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 26857 . . . . . . . . . . . 12  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( _|_ `  ( A  i^i  B ) )  =  ( ( _|_ `  A )  vH  ( _|_ `  B ) ) )
1918adantr 463 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  ( A  e.  CH  /\  C  e.  CH )
)  ->  ( _|_ `  ( A  i^i  B
) )  =  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )
2019ineq1d 3640 . . . . . . . . . 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 2443 . . . . . . . . 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 1285 . . . . . . . 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 3641 . . . . . . 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 463 . . . . . 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 3655 . . . . . . 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 26948 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  C_H  ( _|_ `  B
) ) )
27 cmcm 26946 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  C_H  A ) )
28 choccl 26638 . . . . . . . . . . . . . 14  |-  ( B  e.  CH  ->  ( _|_ `  B )  e. 
CH )
29 cmbr3 26940 . . . . . . . . . . . . . 14  |-  ( ( A  e.  CH  /\  ( _|_ `  B )  e.  CH )  -> 
( A  C_H  ( _|_ `  B )  <->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) ) )
3028, 29sylan2 472 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( A  i^i  ( _|_ `  B ) ) )
33 incom 3632 . . . . . . . . . . 11  |-  ( A  i^i  ( _|_ `  B
) )  =  ( ( _|_ `  B
)  i^i  A )
3432, 33syl6eq 2459 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  B )  i^i  A
) )
35343adantl3 1155 . . . . . . . . 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 715 . . . . . . . 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 3640 . . . . . . 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 2455 . . . . . 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 2443 . . . . 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 3655 . . . . 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 2459 . . . 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 26738 . . . . . . . . . . 11  |-  ( B  e.  CH  ->  ( _|_ `  ( _|_ `  B
) )  =  B )
4342oveq1d 6293 . . . . . . . . . 10  |-  ( B  e.  CH  ->  (
( _|_ `  ( _|_ `  B ) )  vH  C )  =  ( B  vH  C
) )
4443ineq2d 3641 . . . . . . . . 9  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  ( B  vH  C ) ) )
45443ad2ant2 1019 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  ( B  vH  C ) ) )
4645adantr 463 . . . . . . 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 26947 . . . . . . . . . . 11  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  C_H  C  <->  ( _|_ `  B )  C_H  C ) )
48 cmbr3 26940 . . . . . . . . . . . 12  |-  ( ( ( _|_ `  B
)  e.  CH  /\  C  e.  CH )  ->  ( ( _|_ `  B
)  C_H  C  <->  ( ( _|_ `  B )  i^i  ( ( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) ) )
4928, 48sylan 469 . . . . . . . . . . 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 482 . . . . . . . . 9  |-  ( ( ( B  e.  CH  /\  C  e.  CH )  /\  B  C_H  C )  ->  ( ( _|_ `  B )  i^i  (
( _|_ `  ( _|_ `  B ) )  vH  C ) )  =  ( ( _|_ `  B )  i^i  C
) )
52513adantl1 1153 . . . . . . . 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 714 . . . . . . 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 2445 . . . . . 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 3640 . . . . 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 3649 . . . . . . . . 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 3650 . . . . . . . . . . . 12  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
58 inass 3649 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  i^i  ( _|_ `  ( A  i^i  C ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
5957, 58eqtr4i 2434 . . . . . . . . . . 11  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( A  i^i  C
)  i^i  ( _|_ `  ( A  i^i  C
) ) )
60 chocin 26827 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  e.  CH  ->  (
( A  i^i  C
)  i^i  ( _|_ `  ( A  i^i  C
) ) )  =  0H )
612, 60syl 17 . . . . . . . . . . 11  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( ( A  i^i  C )  i^i  ( _|_ `  ( A  i^i  C
) ) )  =  0H )
6259, 61syl5eq 2455 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
6362ineq2d 3641 . . . . . . . . 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 2455 . . . . . . . 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 1016 . . . . . . 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 26823 . . . . . . . . 9  |-  ( ( _|_ `  B )  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6728, 66syl 17 . . . . . . . 8  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
68673ad2ant2 1019 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6965, 68eqtrd 2443 . . . . . 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 463 . . . . 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 2443 . . . 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 2443 . . 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 26768 . . 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 1228 . 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 2410 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    i^i cin 3413    C_ wss 3414   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   SHcsh 26259   CHcch 26260   _|_cort 26261    vH chj 26264   0Hc0h 26266    C_H ccm 26267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cc 8847  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602  ax-hilex 26330  ax-hfvadd 26331  ax-hvcom 26332  ax-hvass 26333  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337  ax-hvmulass 26338  ax-hvdistr1 26339  ax-hvdistr2 26340  ax-hvmul0 26341  ax-hfi 26410  ax-his1 26413  ax-his2 26414  ax-his3 26415  ax-his4 26416  ax-hcompl 26533
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-cn 20021  df-cnp 20022  df-lm 20023  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cfil 21986  df-cau 21987  df-cmet 21988  df-grpo 25607  df-gid 25608  df-ginv 25609  df-gdiv 25610  df-ablo 25698  df-subgo 25718  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-vs 25906  df-nmcv 25907  df-ims 25908  df-dip 26025  df-ssp 26049  df-ph 26142  df-cbn 26193  df-hnorm 26299  df-hba 26300  df-hvsub 26302  df-hlim 26303  df-hcau 26304  df-sh 26538  df-ch 26553  df-oc 26584  df-ch0 26585  df-shs 26640  df-chj 26642  df-cm 26915
This theorem is referenced by:  fh2i  26954  atordi  27716  chirredlem2  27723
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