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Theorem fh2 26210
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 26090 . . . . . . . 8  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  i^i  B
)  e.  CH )
2 chincl 26090 . . . . . . . 8  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( A  i^i  C
)  e.  CH )
3 chjcl 25948 . . . . . . . 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 828 . . . . . 6  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  C  e.  CH )
)  ->  ( ( A  i^i  B )  vH  ( A  i^i  C ) )  e.  CH )
6 chjcl 25948 . . . . . . . 8  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  vH  C
)  e.  CH )
7 chincl 26090 . . . . . . . 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 25815 . . . . . . 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 1192 . . . 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 26131 . . . 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 26120 . . . . . . . . . . 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 26116 . . . . . . . . . . . 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 3699 . . . . . . . . . 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 2508 . . . . . . . . 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 1283 . . . . . . . 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 3700 . . . . . . 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 3714 . . . . . . 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 26207 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  A  C_H  ( _|_ `  B
) ) )
27 cmcm 26205 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_H  B  <->  B  C_H  A ) )
28 choccl 25897 . . . . . . . . . . . . . 14  |-  ( B  e.  CH  ->  ( _|_ `  B )  e. 
CH )
29 cmbr3 26199 . . . . . . . . . . . . . 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 3691 . . . . . . . . . . 11  |-  ( A  i^i  ( _|_ `  B
) )  =  ( ( _|_ `  B
)  i^i  A )
3432, 33syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( A  e.  CH  /\  B  e.  CH )  /\  B  C_H  A )  ->  ( A  i^i  ( ( _|_ `  A
)  vH  ( _|_ `  B ) ) )  =  ( ( _|_ `  B )  i^i  A
) )
35343adantl3 1154 . . . . . . . . 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 3699 . . . . . . 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 2520 . . . . . 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 2508 . . . . 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 3714 . . . . 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 2524 . . . 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 25997 . . . . . . . . . . 11  |-  ( B  e.  CH  ->  ( _|_ `  ( _|_ `  B
) )  =  B )
4342oveq1d 6297 . . . . . . . . . 10  |-  ( B  e.  CH  ->  (
( _|_ `  ( _|_ `  B ) )  vH  C )  =  ( B  vH  C
) )
4443ineq2d 3700 . . . . . . . . 9  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  ( ( _|_ `  ( _|_ `  B
) )  vH  C
) )  =  ( ( _|_ `  B
)  i^i  ( B  vH  C ) ) )
45443ad2ant2 1018 . . . . . . . 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 26206 . . . . . . . . . . 11  |-  ( ( B  e.  CH  /\  C  e.  CH )  ->  ( B  C_H  C  <->  ( _|_ `  B )  C_H  C ) )
48 cmbr3 26199 . . . . . . . . . . . 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 1152 . . . . . . . 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 2510 . . . . . 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 3699 . . . . 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 3708 . . . . . . . . 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 3709 . . . . . . . . . . . 12  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
58 inass 3708 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  i^i  ( _|_ `  ( A  i^i  C ) ) )  =  ( A  i^i  ( C  i^i  ( _|_ `  ( A  i^i  C ) ) ) )
5957, 58eqtr4i 2499 . . . . . . . . . . 11  |-  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  ( ( A  i^i  C
)  i^i  ( _|_ `  ( A  i^i  C
) ) )
60 chocin 26086 . . . . . . . . . . . 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 2520 . . . . . . . . . 10  |-  ( ( A  e.  CH  /\  C  e.  CH )  ->  ( C  i^i  ( A  i^i  ( _|_ `  ( A  i^i  C ) ) ) )  =  0H )
6362ineq2d 3700 . . . . . . . . 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 2520 . . . . . . . 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 1015 . . . . . . 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 26082 . . . . . . . . 9  |-  ( ( _|_ `  B )  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6728, 66syl 16 . . . . . . . 8  |-  ( B  e.  CH  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
68673ad2ant2 1018 . . . . . . 7  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  C  e.  CH )  ->  (
( _|_ `  B
)  i^i  0H )  =  0H )
6965, 68eqtrd 2508 . . . . . 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 2508 . . . 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 2508 . . 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 26027 . . 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 1226 . 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 2475 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 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   SHcsh 25518   CHcch 25519   _|_cort 25520    vH chj 25523   0Hc0h 25525    C_H ccm 25526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25589  ax-hfvadd 25590  ax-hvcom 25591  ax-hvass 25592  ax-hv0cl 25593  ax-hvaddid 25594  ax-hfvmul 25595  ax-hvmulid 25596  ax-hvmulass 25597  ax-hvdistr1 25598  ax-hvdistr2 25599  ax-hvmul0 25600  ax-hfi 25669  ax-his1 25672  ax-his2 25673  ax-his3 25674  ax-his4 25675  ax-hcompl 25792
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-rlim 13268  df-sum 13465  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-cn 19491  df-cnp 19492  df-lm 19493  df-haus 19579  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cfil 21426  df-cau 21427  df-cmet 21428  df-grpo 24866  df-gid 24867  df-ginv 24868  df-gdiv 24869  df-ablo 24957  df-subgo 24977  df-vc 25112  df-nv 25158  df-va 25161  df-ba 25162  df-sm 25163  df-0v 25164  df-vs 25165  df-nmcv 25166  df-ims 25167  df-dip 25284  df-ssp 25308  df-ph 25401  df-cbn 25452  df-hnorm 25558  df-hba 25559  df-hvsub 25561  df-hlim 25562  df-hcau 25563  df-sh 25797  df-ch 25812  df-oc 25843  df-ch0 25844  df-shs 25899  df-chj 25901  df-cm 26174
This theorem is referenced by:  fh2i  26213  atordi  26976  chirredlem2  26983
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