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Theorem mdexchi 28000
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 27198 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  (
( C  vH  A
)  vH  x )  =  ( C  vH  ( A  vH  x
) ) )
41, 2, 3mp3an12 1356 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
( C  vH  A
)  vH  x )  =  ( C  vH  ( A  vH  x
) ) )
51, 2chjcli 27122 . . . . . . . . . . . . . . 15  |-  ( C  vH  A )  e. 
CH
6 chjcom 27171 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CH  /\  ( C  vH  A )  e.  CH )  -> 
( x  vH  ( C  vH  A ) )  =  ( ( C  vH  A )  vH  x ) )
75, 6mpan2 678 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
x  vH  ( C  vH  A ) )  =  ( ( C  vH  A )  vH  x
) )
8 chjcl 27022 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  CH  /\  x  e.  CH )  ->  ( A  vH  x
)  e.  CH )
92, 8mpan 677 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  ( A  vH  x )  e. 
CH )
10 chjcom 27171 . . . . . . . . . . . . . . 15  |-  ( ( ( A  vH  x
)  e.  CH  /\  C  e.  CH )  ->  ( ( A  vH  x )  vH  C
)  =  ( C  vH  ( A  vH  x ) ) )
119, 1, 10sylancl 669 . . . . . . . . . . . . . 14  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  C )  =  ( C  vH  ( A  vH  x
) ) )
124, 7, 113eqtr4d 2497 . . . . . . . . . . . . 13  |-  ( x  e.  CH  ->  (
x  vH  ( C  vH  A ) )  =  ( ( A  vH  x )  vH  C
) )
1312ineq1d 3635 . . . . . . . . . . . 12  |-  ( x  e.  CH  ->  (
( x  vH  ( C  vH  A ) )  i^i  B )  =  ( ( ( A  vH  x )  vH  C )  i^i  B
) )
14 inass 3644 . . . . . . . . . . . . 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 3627 . . . . . . . . . . . . . . 15  |-  ( ( A  vH  B )  i^i  B )  =  ( B  i^i  ( A  vH  B ) )
16 mdexch.2 . . . . . . . . . . . . . . . . . 18  |-  B  e. 
CH
172, 16chjcomi 27133 . . . . . . . . . . . . . . . . 17  |-  ( A  vH  B )  =  ( B  vH  A
)
1817ineq2i 3633 . . . . . . . . . . . . . . . 16  |-  ( B  i^i  ( A  vH  B ) )  =  ( B  i^i  ( B  vH  A ) )
1916, 2chabs2i 27184 . . . . . . . . . . . . . . . 16  |-  ( B  i^i  ( B  vH  A ) )  =  B
2018, 19eqtri 2475 . . . . . . . . . . . . . . 15  |-  ( B  i^i  ( A  vH  B ) )  =  B
2115, 20eqtri 2475 . . . . . . . . . . . . . 14  |-  ( ( A  vH  B )  i^i  B )  =  B
2221ineq2i 3633 . . . . . . . . . . . . 13  |-  ( ( ( A  vH  x
)  vH  C )  i^i  ( ( A  vH  B )  i^i  B
) )  =  ( ( ( A  vH  x )  vH  C
)  i^i  B )
2314, 22eqtri 2475 . . . . . . . . . . . 12  |-  ( ( ( ( A  vH  x )  vH  C
)  i^i  ( A  vH  B ) )  i^i 
B )  =  ( ( ( A  vH  x )  vH  C
)  i^i  B )
2413, 23syl6eqr 2505 . . . . . . . . . . 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 733 . . . . . . . . . 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 27176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  CH  /\  B  e.  CH  /\  A  e.  CH )  /\  x  C_  B )  ->  ( A  vH  x )  C_  ( A  vH  B ) )
2726ex 436 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  CH  /\  B  e.  CH  /\  A  e.  CH )  ->  (
x  C_  B  ->  ( A  vH  x ) 
C_  ( A  vH  B ) ) )
2816, 2, 27mp3an23 1358 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( A  vH  x ) 
C_  ( A  vH  B ) ) )
292, 16chjcli 27122 . . . . . . . . . . . . . . . . . 18  |-  ( A  vH  B )  e. 
CH
30 mdi 27960 . . . . . . . . . . . . . . . . . . 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 610 . . . . . . . . . . . . . . . . . 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 1356 . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 27125 . . . . . . . . . . . . . . . . 17  |-  ( C  i^i  ( A  vH  B ) )  e. 
CH
39 chlej2 27176 . . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . . 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 1356 . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . 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 27171 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  vH  x
)  e.  CH  /\  A  e.  CH )  ->  ( ( A  vH  x )  vH  A
)  =  ( A  vH  ( A  vH  x ) ) )
459, 2, 44sylancl 669 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  A )  =  ( A  vH  ( A  vH  x
) ) )
462chjidmi 27186 . . . . . . . . . . . . . . . . . 18  |-  ( A  vH  A )  =  A
4746oveq1i 6305 . . . . . . . . . . . . . . . . 17  |-  ( ( A  vH  A )  vH  x )  =  ( A  vH  x
)
48 chjass 27198 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CH  /\  A  e.  CH  /\  x  e.  CH )  ->  (
( A  vH  A
)  vH  x )  =  ( A  vH  ( A  vH  x
) ) )
492, 2, 48mp3an12 1356 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CH  ->  (
( A  vH  A
)  vH  x )  =  ( A  vH  ( A  vH  x
) ) )
50 chjcom 27171 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  CH  /\  x  e.  CH )  ->  ( A  vH  x
)  =  ( x  vH  A ) )
512, 50mpan 677 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  CH  ->  ( A  vH  x )  =  ( x  vH  A
) )
5247, 49, 513eqtr3a 2511 . . . . . . . . . . . . . . . 16  |-  ( x  e.  CH  ->  ( A  vH  ( A  vH  x ) )  =  ( x  vH  A
) )
5345, 52eqtrd 2487 . . . . . . . . . . . . . . 15  |-  ( x  e.  CH  ->  (
( A  vH  x
)  vH  A )  =  ( x  vH  A ) )
5453adantr 467 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CH  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  x )  vH  A
)  =  ( x  vH  A ) )
5543, 54sseqtrd 3470 . . . . . . . . . . . . 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 756 . . . . . . . . . . . 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 3468 . . . . . . . . . . 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 3659 . . . . . . . . . . 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 17 . . . . . . . . . 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 3468 . . . . . . . . 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 723 . . . . . . . 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 27960 . . . . . . . . . . . . . 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 610 . . . . . . . . . . . . 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 1356 . . . . . . . . . . . 12  |-  ( x  e.  CH  ->  ( A  MH  B  ->  ( x  C_  B  ->  ( ( x  vH  A
)  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6564com23 81 . . . . . . . . . . 11  |-  ( x  e.  CH  ->  (
x  C_  B  ->  ( A  MH  B  -> 
( ( x  vH  A )  i^i  B
)  =  ( x  vH  ( A  i^i  B ) ) ) ) )
6665imp31 434 . . . . . . . . . 10  |-  ( ( ( x  e.  CH  /\  x  C_  B )  /\  A  MH  B
)  ->  ( (
x  vH  A )  i^i  B )  =  ( x  vH  ( A  i^i  B ) ) )
672, 1chub2i 27135 . . . . . . . . . . . . 13  |-  A  C_  ( C  vH  A )
68 ssrin 3659 . . . . . . . . . . . . 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 27125 . . . . . . . . . . . . 13  |-  ( A  i^i  B )  e. 
CH
715, 16chincli 27125 . . . . . . . . . . . . 13  |-  ( ( C  vH  A )  i^i  B )  e. 
CH
72 chlej2 27176 . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . 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 1356 . . . . . . . . . . . 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 20 . . . . . . . . . . 11  |-  ( x  e.  CH  ->  (
x  vH  ( A  i^i  B ) )  C_  ( x  vH  (
( C  vH  A
)  i^i  B )
) )
7675ad2antrr 733 . . . . . . . . . 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 3468 . . . . . . . . 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 724 . . . . . . . 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 3444 . . . . . . 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 609 . . . . . 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 82 . . . . 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 1205 . . . 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 2802 . . 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 27961 . . . 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 679 . . 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 216 . 2  |-  ( ( A  MH  B  /\  C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) )  C_  A )  ->  ( C  vH  A )  MH  B )
871, 2chjcomi 27133 . . . . 5  |-  ( C  vH  A )  =  ( A  vH  C
)
88 incom 3627 . . . . . 6  |-  ( B  i^i  ( A  vH  B ) )  =  ( ( A  vH  B )  i^i  B
)
8918, 88, 193eqtr3ri 2484 . . . . 5  |-  B  =  ( ( A  vH  B )  i^i  B
)
9087, 89ineq12i 3634 . . . 4  |-  ( ( C  vH  A )  i^i  B )  =  ( ( A  vH  C )  i^i  (
( A  vH  B
)  i^i  B )
)
91 inass 3644 . . . . 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 27134 . . . . . . . 8  |-  A  C_  ( A  vH  B )
93 mdi 27960 . . . . . . . . . 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 610 . . . . . . . . 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 1366 . . . . . . . 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 20 . . . . . . 7  |-  ( C  MH  ( A  vH  B )  ->  (
( A  vH  C
)  i^i  ( A  vH  B ) )  =  ( A  vH  ( C  i^i  ( A  vH  B ) ) ) )
972, 38chjcomi 27133 . . . . . . . 8  |-  ( A  vH  ( C  i^i  ( A  vH  B ) ) )  =  ( ( C  i^i  ( A  vH  B ) )  vH  A )
9838, 2chlejb1i 27141 . . . . . . . . 9  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  <->  ( ( C  i^i  ( A  vH  B ) )  vH  A )  =  A )
9998biimpi 198 . . . . . . . 8  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  ->  (
( C  i^i  ( A  vH  B ) )  vH  A )  =  A )
10097, 99syl5eq 2499 . . . . . . 7  |-  ( ( C  i^i  ( A  vH  B ) ) 
C_  A  ->  ( A  vH  ( C  i^i  ( A  vH  B ) ) )  =  A )
10196, 100sylan9eq 2507 . . . . . 6  |-  ( ( C  MH  ( A  vH  B )  /\  ( C  i^i  ( A  vH  B ) ) 
C_  A )  -> 
( ( A  vH  C )  i^i  ( A  vH  B ) )  =  A )
102101ineq1d 3635 . . . . 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 2501 . . . 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 2499 . . 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 1027 . 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 535 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739    i^i cin 3405    C_ wss 3406   class class class wbr 4405  (class class class)co 6295   CHcch 26594    vH chj 26598    MH cmd 26631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cc 8870  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624  ax-hilex 26664  ax-hfvadd 26665  ax-hvcom 26666  ax-hvass 26667  ax-hv0cl 26668  ax-hvaddid 26669  ax-hfvmul 26670  ax-hvmulid 26671  ax-hvmulass 26672  ax-hvdistr1 26673  ax-hvdistr2 26674  ax-hvmul0 26675  ax-hfi 26744  ax-his1 26747  ax-his2 26748  ax-his3 26749  ax-his4 26750  ax-hcompl 26867
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-map 7479  df-pm 7480  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-acn 8381  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-fbas 18979  df-fg 18980  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-cn 20255  df-cnp 20256  df-lm 20257  df-haus 20343  df-tx 20589  df-hmeo 20782  df-fil 20873  df-fm 20965  df-flim 20966  df-flf 20967  df-xms 21347  df-ms 21348  df-tms 21349  df-cfil 22237  df-cau 22238  df-cmet 22239  df-grpo 25931  df-gid 25932  df-ginv 25933  df-gdiv 25934  df-ablo 26022  df-subgo 26042  df-vc 26177  df-nv 26223  df-va 26226  df-ba 26227  df-sm 26228  df-0v 26229  df-vs 26230  df-nmcv 26231  df-ims 26232  df-dip 26349  df-ssp 26373  df-ph 26466  df-cbn 26517  df-hnorm 26633  df-hba 26634  df-hvsub 26636  df-hlim 26637  df-hcau 26638  df-sh 26872  df-ch 26886  df-oc 26917  df-ch0 26918  df-shs 26973  df-chj 26975  df-md 27945
This theorem is referenced by: (None)
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