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Theorem mdexchi 26930
Description: An exchange lemma for modular pairs. Lemma 1.6 of [MaedaMaeda] p. 2. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdexch.1  |-  A  e. 
CH
mdexch.2  |-  B  e. 
CH
mdexch.3  |-  C  e. 
CH
Assertion
Ref Expression
mdexchi  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  (
( C  vH  A
)  MH  B  /\  ( ( C  vH  A )  i^i  B
)  =  ( A  i^i  B ) ) )

Proof of Theorem mdexchi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mdexch.3 . . . . . . . . . . . . . . 15  |-  C  e. 
CH
2 mdexch.1 . . . . . . . . . . . . . . 15  |-  A  e. 
CH
3 chjass 26127 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  (
( C  vH  A
)  vH  x )  =  ( C  vH  ( A  vH  x
) ) )
41, 2, 3mp3an12 1314 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
( C  vH  A
)  vH  x )  =  ( C  vH  ( A  vH  x
) ) )
51, 2chjcli 26051 . . . . . . . . . . . . . . 15  |-  ( C  vH  A )  e. 
CH
6 chjcom 26100 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CH  /\  ( C  vH  A )  e.  CH )  -> 
( x  vH  ( C  vH  A ) )  =  ( ( C  vH  A )  vH  x ) )
75, 6mpan2 671 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
x  vH  ( C  vH  A ) )  =  ( ( C  vH  A )  vH  x
) )
8 chjcl 25951 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  CH )  ->  ( A  vH  x
)  e.  CH )
92, 8mpan 670 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  ( A  vH  x )  e. 
CH )
10 chjcom 26100 . . . . . . . . . . . . . . 15  |-  ( ( ( A  vH  x
)  e.  CH  /\  C  e.  CH )  ->  ( ( A  vH  x )  vH  C
)  =  ( C  vH  ( A  vH  x ) ) )
119, 1, 10sylancl 662 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  C )  =  ( C  vH  ( A  vH  x
) ) )
124, 7, 113eqtr4d 2518 . . . . . . . . . . . . 13  |-  ( x  e.  CH  ->  (
x  vH  ( C  vH  A ) )  =  ( ( A  vH  x )  vH  C
) )
1312ineq1d 3699 . . . . . . . . . . . 12  |-  ( x  e.  CH  ->  (
( x  vH  ( C  vH  A ) )  i^i  B )  =  ( ( ( A  vH  x )  vH  C )  i^i  B
) )
14 inass 3708 . . . . . . . . . . . . 13  |-  ( ( ( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  i^i 
B )  =  ( ( ( A  vH  x )  vH  C
)  i^i  ( ( A  vH  B )  i^i 
B ) )
15 incom 3691 . . . . . . . . . . . . . . 15  |-  ( ( A  vH  B )  i^i  B )  =  ( B  i^i  ( A  vH  B ) )
16 mdexch.2 . . . . . . . . . . . . . . . . . 18  |-  B  e. 
CH
172, 16chjcomi 26062 . . . . . . . . . . . . . . . . 17  |-  ( A  vH  B )  =  ( B  vH  A
)
1817ineq2i 3697 . . . . . . . . . . . . . . . 16  |-  ( B  i^i  ( A  vH  B ) )  =  ( B  i^i  ( B  vH  A ) )
1916, 2chabs2i 26113 . . . . . . . . . . . . . . . 16  |-  ( B  i^i  ( B  vH  A ) )  =  B
2018, 19eqtri 2496 . . . . . . . . . . . . . . 15  |-  ( B  i^i  ( A  vH  B ) )  =  B
2115, 20eqtri 2496 . . . . . . . . . . . . . 14  |-  ( ( A  vH  B )  i^i  B )  =  B
2221ineq2i 3697 . . . . . . . . . . . . 13  |-  ( ( ( A  vH  x
)  vH  C )  i^i  ( ( A  vH  B )  i^i  B
) )  =  ( ( ( A  vH  x )  vH  C
)  i^i  B )
2314, 22eqtri 2496 . . . . . . . . . . . 12  |-  ( ( ( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  i^i 
B )  =  ( ( ( A  vH  x )  vH  C
)  i^i  B )
2413, 23syl6eqr 2526 . . . . . . . . . . 11  |-  ( x  e.  CH  ->  (
( x  vH  ( C  vH  A ) )  i^i  B )  =  ( ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B
) )  i^i  B
) )
2524ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  =  ( ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B ) )  i^i  B ) )
26 chlej2 26105 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  CH  /\  B  e.  CH  /\  A  e.  CH )  /\  x  C_  B )  ->  ( A  vH  x )  C_  ( A  vH  B ) )
2726ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CH  /\  B  e.  CH  /\  A  e.  CH )  ->  (
x  C_  B  ->  ( A  vH  x ) 
C_  ( A  vH  B ) ) )
2816, 2, 27mp3an23 1316 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( A  vH  x ) 
C_  ( A  vH  B ) ) )
292, 16chjcli 26051 . . . . . . . . . . . . . . . . . 18  |-  ( A  vH  B )  e. 
CH
30 mdi 26890 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( C  e.  CH  /\  ( A  vH  B
)  e.  CH  /\  ( A  vH  x
)  e.  CH )  /\  ( C  MH  ( A  vH  B )  /\  ( A  vH  x
)  C_  ( A  vH  B ) ) )  ->  ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B
) )  =  ( ( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) ) )
3130exp32 605 . . . . . . . . . . . . . . . . . 18  |-  ( ( C  e.  CH  /\  ( A  vH  B )  e.  CH  /\  ( A  vH  x )  e. 
CH )  ->  ( C  MH  ( A  vH  B )  ->  (
( A  vH  x
)  C_  ( A  vH  B )  ->  (
( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  =  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
321, 29, 31mp3an12 1314 . . . . . . . . . . . . . . . . 17  |-  ( ( A  vH  x )  e.  CH  ->  ( C  MH  ( A  vH  B )  ->  (
( A  vH  x
)  C_  ( A  vH  B )  ->  (
( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  =  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
339, 32syl 16 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  ( C  MH  ( A  vH  B )  ->  (
( A  vH  x
)  C_  ( A  vH  B )  ->  (
( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  =  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
3433com23 78 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
( A  vH  x
)  C_  ( A  vH  B )  ->  ( C  MH  ( A  vH  B )  ->  (
( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  =  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
3528, 34syld 44 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( C  MH  ( A  vH  B )  -> 
( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B ) )  =  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
3635imp31 432 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  C  MH  ( A  vH  B ) )  ->  ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B
) )  =  ( ( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) ) )
3736adantrr 716 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B
) )  =  ( ( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) ) )
381, 29chincli 26054 . . . . . . . . . . . . . . . . 17  |-  ( C  i^i  ( A  vH  B ) )  e. 
CH
39 chlej2 26105 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( C  i^i  ( A  vH  B ) )  e.  CH  /\  A  e.  CH  /\  ( A  vH  x )  e. 
CH )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  (
( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) )  C_  ( ( A  vH  x )  vH  A
) )
4039ex 434 . . . . . . . . . . . . . . . . 17  |-  ( ( ( C  i^i  ( A  vH  B ) )  e.  CH  /\  A  e.  CH  /\  ( A  vH  x )  e. 
CH )  ->  (
( C  i^i  ( A  vH  B ) ) 
C_  A  ->  (
( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) )  C_  ( ( A  vH  x )  vH  A
) ) )
4138, 2, 40mp3an12 1314 . . . . . . . . . . . . . . . 16  |-  ( ( A  vH  x )  e.  CH  ->  (
( C  i^i  ( A  vH  B ) ) 
C_  A  ->  (
( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) )  C_  ( ( A  vH  x )  vH  A
) ) )
429, 41syl 16 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
( C  i^i  ( A  vH  B ) ) 
C_  A  ->  (
( A  vH  x
)  vH  ( C  i^i  ( A  vH  B
) ) )  C_  ( ( A  vH  x )  vH  A
) ) )
4342imp 429 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) 
C_  ( ( A  vH  x )  vH  A ) )
44 chjcom 26100 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  vH  x
)  e.  CH  /\  A  e.  CH )  ->  ( ( A  vH  x )  vH  A
)  =  ( A  vH  ( A  vH  x ) ) )
459, 2, 44sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  A )  =  ( A  vH  ( A  vH  x
) ) )
462chjidmi 26115 . . . . . . . . . . . . . . . . . 18  |-  ( A  vH  A )  =  A
4746oveq1i 6292 . . . . . . . . . . . . . . . . 17  |-  ( ( A  vH  A )  vH  x )  =  ( A  vH  x
)
48 chjass 26127 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  (
( A  vH  A
)  vH  x )  =  ( A  vH  ( A  vH  x
) ) )
492, 2, 48mp3an12 1314 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CH  ->  (
( A  vH  A
)  vH  x )  =  ( A  vH  ( A  vH  x
) ) )
50 chjcom 26100 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CH  /\  x  e.  CH )  ->  ( A  vH  x
)  =  ( x  vH  A ) )
512, 50mpan 670 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CH  ->  ( A  vH  x )  =  ( x  vH  A
) )
5247, 49, 513eqtr3a 2532 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  ( A  vH  ( A  vH  x ) )  =  ( x  vH  A
) )
5345, 52eqtrd 2508 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  A )  =  ( x  vH  A ) )
5453adantr 465 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  x )  vH  A
)  =  ( x  vH  A ) )
5543, 54sseqtrd 3540 . . . . . . . . . . . . 13  |-  ( ( x  e.  CH  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) ) 
C_  ( x  vH  A ) )
5655ad2ant2rl 748 . . . . . . . . . . . 12  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( A  vH  x )  vH  ( C  i^i  ( A  vH  B ) ) )  C_  ( x  vH  A ) )
5737, 56eqsstrd 3538 . . . . . . . . . . 11  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( ( A  vH  x )  vH  C )  i^i  ( A  vH  B
) )  C_  (
x  vH  A )
)
58 ssrin 3723 . . . . . . . . . . 11  |-  ( ( ( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  C_  ( x  vH  A )  ->  ( ( ( ( A  vH  x
)  vH  C )  i^i  ( A  vH  B
) )  i^i  B
)  C_  ( (
x  vH  A )  i^i  B ) )
5957, 58syl 16 . . . . . . . . . 10  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( ( ( A  vH  x
)  vH  C )  i^i  ( A  vH  B
) )  i^i  B
)  C_  ( (
x  vH  A )  i^i  B ) )
6025, 59eqsstrd 3538 . . . . . . . . 9  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
( x  vH  A
)  i^i  B )
)
6160adantrl 715 . . . . . . . 8  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( A  MH  B  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) ) )  ->  ( (
x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
( x  vH  A
)  i^i  B )
)
62 mdi 26890 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  x  e.  CH )  /\  ( A  MH  B  /\  x  C_  B ) )  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
6362exp32 605 . . . . . . . . . . . . 13  |-  ( ( A  e.  CH  /\  B  e.  CH  /\  x  e.  CH )  ->  ( A  MH  B  ->  ( x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
642, 16, 63mp3an12 1314 . . . . . . . . . . . 12  |-  ( x  e.  CH  ->  ( A  MH  B  ->  ( x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6564com23 78 . . . . . . . . . . 11  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( A  MH  B  -> 
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6665imp31 432 . . . . . . . . . 10  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  A  MH  B
)  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
672, 1chub2i 26064 . . . . . . . . . . . . 13  |-  A  C_  ( C  vH  A )
68 ssrin 3723 . . . . . . . . . . . . 13  |-  ( A 
C_  ( C  vH  A )  ->  ( A  i^i  B )  C_  ( ( C  vH  A )  i^i  B
) )
6967, 68ax-mp 5 . . . . . . . . . . . 12  |-  ( A  i^i  B )  C_  ( ( C  vH  A )  i^i  B
)
702, 16chincli 26054 . . . . . . . . . . . . 13  |-  ( A  i^i  B )  e. 
CH
715, 16chincli 26054 . . . . . . . . . . . . 13  |-  ( ( C  vH  A )  i^i  B )  e. 
CH
72 chlej2 26105 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  i^i  B )  e.  CH  /\  ( ( C  vH  A )  i^i  B
)  e.  CH  /\  x  e.  CH )  /\  ( A  i^i  B
)  C_  ( ( C  vH  A )  i^i 
B ) )  -> 
( x  vH  ( A  i^i  B ) ) 
C_  ( x  vH  ( ( C  vH  A )  i^i  B
) ) )
7372ex 434 . . . . . . . . . . . . 13  |-  ( ( ( A  i^i  B
)  e.  CH  /\  ( ( C  vH  A )  i^i  B
)  e.  CH  /\  x  e.  CH )  ->  ( ( A  i^i  B )  C_  ( ( C  vH  A )  i^i 
B )  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) ) )
7470, 71, 73mp3an12 1314 . . . . . . . . . . . 12  |-  ( x  e.  CH  ->  (
( A  i^i  B
)  C_  ( ( C  vH  A )  i^i 
B )  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) ) )
7569, 74mpi 17 . . . . . . . . . . 11  |-  ( x  e.  CH  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) )
7675ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  A  MH  B
)  ->  ( x  vH  ( A  i^i  B
) )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) )
7766, 76eqsstrd 3538 . . . . . . . . 9  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  A  MH  B
)  ->  ( (
x  vH  A )  i^i  B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) )
7877adantrr 716 . . . . . . . 8  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( A  MH  B  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) ) )  ->  ( (
x  vH  A )  i^i  B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) )
7961, 78sstrd 3514 . . . . . . 7  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  ( A  MH  B  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) ) )  ->  ( (
x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) )
8079exp31 604 . . . . . 6  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( ( A  MH  B  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) ) ) )
8180com3r 79 . . . . 5  |-  ( ( A  MH  B  /\  ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A ) )  ->  ( x  e. 
CH  ->  ( x  C_  B  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) ) ) )
82813impb 1192 . . . 4  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  (
x  e.  CH  ->  ( x  C_  B  ->  ( ( x  vH  ( C  vH  A ) )  i^i  B )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) ) ) )
8382ralrimiv 2876 . . 3  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) ) )
84 mdbr2 26891 . . . 4  |-  ( ( ( C  vH  A
)  e.  CH  /\  B  e.  CH )  ->  ( ( C  vH  A )  MH  B  <->  A. x  e.  CH  (
x  C_  B  ->  ( ( x  vH  ( C  vH  A ) )  i^i  B )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) ) ) )
855, 16, 84mp2an 672 . . 3  |-  ( ( C  vH  A )  MH  B  <->  A. x  e.  CH  ( x  C_  B  ->  ( ( x  vH  ( C  vH  A ) )  i^i 
B )  C_  (
x  vH  ( ( C  vH  A )  i^i 
B ) ) ) )
8683, 85sylibr 212 . 2  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  ( C  vH  A )  MH  B )
871, 2chjcomi 26062 . . . . 5  |-  ( C  vH  A )  =  ( A  vH  C
)
88 incom 3691 . . . . . 6  |-  ( B  i^i  ( A  vH  B ) )  =  ( ( A  vH  B )  i^i  B
)
8918, 88, 193eqtr3ri 2505 . . . . 5  |-  B  =  ( ( A  vH  B )  i^i  B
)
9087, 89ineq12i 3698 . . . 4  |-  ( ( C  vH  A )  i^i  B )  =  ( ( A  vH  C )  i^i  (
( A  vH  B
)  i^i  B )
)
91 inass 3708 . . . . 5  |-  ( ( ( A  vH  C
)  i^i  ( A  vH  B ) )  i^i 
B )  =  ( ( A  vH  C
)  i^i  ( ( A  vH  B )  i^i 
B ) )
922, 16chub1i 26063 . . . . . . . 8  |-  A  C_  ( A  vH  B )
93 mdi 26890 . . . . . . . . . 10  |-  ( ( ( C  e.  CH  /\  ( A  vH  B
)  e.  CH  /\  A  e.  CH )  /\  ( C  MH  ( A  vH  B )  /\  A  C_  ( A  vH  B ) ) )  ->  ( ( A  vH  C )  i^i  ( A  vH  B
) )  =  ( A  vH  ( C  i^i  ( A  vH  B ) ) ) )
9493exp32 605 . . . . . . . . 9  |-  ( ( C  e.  CH  /\  ( A  vH  B )  e.  CH  /\  A  e.  CH )  ->  ( C  MH  ( A  vH  B )  ->  ( A  C_  ( A  vH  B )  ->  (
( A  vH  C
)  i^i  ( A  vH  B ) )  =  ( A  vH  ( C  i^i  ( A  vH  B ) ) ) ) ) )
951, 29, 2, 94mp3an 1324 . . . . . . . 8  |-  ( C  MH  ( A  vH  B )  ->  ( A  C_  ( A  vH  B )  ->  (
( A  vH  C
)  i^i  ( A  vH  B ) )  =  ( A  vH  ( C  i^i  ( A  vH  B ) ) ) ) )
9692, 95mpi 17 . . . . . . 7  |-  ( C  MH  ( A  vH  B )  ->  (
( A  vH  C
)  i^i  ( A  vH  B ) )  =  ( A  vH  ( C  i^i  ( A  vH  B ) ) ) )
972, 38chjcomi 26062 . . . . . . . 8  |-  ( A  vH  ( C  i^i  ( A  vH  B ) ) )  =  ( ( C  i^i  ( A  vH  B ) )  vH  A )
9838, 2chlejb1i 26070 . . . . . . . . 9  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  <->  ( ( C  i^i  ( A  vH  B ) )  vH  A )  =  A )
9998biimpi 194 . . . . . . . 8  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  ->  (
( C  i^i  ( A  vH  B ) )  vH  A )  =  A )
10097, 99syl5eq 2520 . . . . . . 7  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  ->  ( A  vH  ( C  i^i  ( A  vH  B ) ) )  =  A )
10196, 100sylan9eq 2528 . . . . . 6  |-  ( ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  C )  i^i  ( A  vH  B ) )  =  A )
102101ineq1d 3699 . . . . 5  |-  ( ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( ( A  vH  C )  i^i  ( A  vH  B
) )  i^i  B
)  =  ( A  i^i  B ) )
10391, 102syl5eqr 2522 . . . 4  |-  ( ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  C )  i^i  (
( A  vH  B
)  i^i  B )
)  =  ( A  i^i  B ) )
10490, 103syl5eq 2520 . . 3  |-  ( ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( C  vH  A )  i^i  B
)  =  ( A  i^i  B ) )
1051043adant1 1014 . 2  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  (
( C  vH  A
)  i^i  B )  =  ( A  i^i  B ) )
10686, 105jca 532 1  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  (
( C  vH  A
)  MH  B  /\  ( ( C  vH  A )  i^i  B
)  =  ( A  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   class class class wbr 4447  (class class class)co 6282   CHcch 25522    vH chj 25526    MH cmd 25559
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 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795
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 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-shs 25902  df-chj 25904  df-md 26875
This theorem is referenced by: (None)
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