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Theorem bcseqi 22575
Description: Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 22635. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem7t.1  |-  A  e. 
~H
normlem7t.2  |-  B  e. 
~H
Assertion
Ref Expression
bcseqi  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  <->  ( ( B  .ih  B )  .h  A )  =  ( ( A  .ih  B
)  .h  B ) )

Proof of Theorem bcseqi
StepHypRef Expression
1 normlem7t.2 . . . . . . . 8  |-  B  e. 
~H
21, 1hicli 22536 . . . . . . 7  |-  ( B 
.ih  B )  e.  CC
3 normlem7t.1 . . . . . . 7  |-  A  e. 
~H
42, 3hvmulcli 22470 . . . . . 6  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 22536 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 22470 . . . . . 6  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
74, 6, 4, 6normlem9 22573 . . . . 5  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  ( ( ( ( ( B  .ih  B )  .h  A )  .ih  ( ( B  .ih  B )  .h  A ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) ) )  -  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) ) )
8 oveq1 6047 . . . . . . . . . 10  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( A  .ih  B )  x.  ( B 
.ih  A ) )  x.  ( B  .ih  B ) )  =  ( ( ( A  .ih  A )  x.  ( B 
.ih  B ) )  x.  ( B  .ih  B ) ) )
98eqcomd 2409 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( A  .ih  A )  x.  ( B 
.ih  B ) )  x.  ( B  .ih  B ) )  =  ( ( ( A  .ih  B )  x.  ( B 
.ih  A ) )  x.  ( B  .ih  B ) ) )
10 his5 22541 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  e.  CC  /\  ( ( B  .ih  B )  .h  A )  e.  ~H  /\  A  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( * `  ( B  .ih  B ) )  x.  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A ) ) )
112, 4, 3, 10mp3an 1279 . . . . . . . . . 10  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( * `  ( B  .ih  B ) )  x.  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A ) )
12 hiidrcl 22550 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
13 cjre 11899 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  e.  RR  ->  (
* `  ( B  .ih  B ) )  =  ( B  .ih  B
) )
141, 12, 13mp2b 10 . . . . . . . . . . 11  |-  ( * `
 ( B  .ih  B ) )  =  ( B  .ih  B )
15 ax-his3 22539 . . . . . . . . . . . 12  |-  ( ( ( B  .ih  B
)  e.  CC  /\  A  e.  ~H  /\  A  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) ) )
162, 3, 3, 15mp3an 1279 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
1714, 16oveq12i 6052 . . . . . . . . . 10  |-  ( ( * `  ( B 
.ih  B ) )  x.  ( ( ( B  .ih  B )  .h  A )  .ih  A ) )  =  ( ( B  .ih  B
)  x.  ( ( B  .ih  B )  x.  ( A  .ih  A ) ) )
183, 3hicli 22536 . . . . . . . . . . . . 13  |-  ( A 
.ih  A )  e.  CC
192, 18mulcli 9051 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  e.  CC
202, 19mulcomi 9052 . . . . . . . . . . 11  |-  ( ( B  .ih  B )  x.  ( ( B 
.ih  B )  x.  ( A  .ih  A
) ) )  =  ( ( ( B 
.ih  B )  x.  ( A  .ih  A
) )  x.  ( B  .ih  B ) )
2118, 2mulcomi 9052 . . . . . . . . . . . 12  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  x.  ( A 
.ih  A ) )
2221oveq1i 6050 . . . . . . . . . . 11  |-  ( ( ( A  .ih  A
)  x.  ( B 
.ih  B ) )  x.  ( B  .ih  B ) )  =  ( ( ( B  .ih  B )  x.  ( A 
.ih  A ) )  x.  ( B  .ih  B ) )
2320, 22eqtr4i 2427 . . . . . . . . . 10  |-  ( ( B  .ih  B )  x.  ( ( B 
.ih  B )  x.  ( A  .ih  A
) ) )  =  ( ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  x.  ( B  .ih  B ) )
2411, 17, 233eqtri 2428 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  x.  ( B  .ih  B ) )
25 his5 22541 . . . . . . . . . . 11  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( ( B  .ih  B )  .h  A )  e.  ~H  /\  B  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( * `  ( A  .ih  B ) )  x.  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B ) ) )
265, 4, 1, 25mp3an 1279 . . . . . . . . . 10  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( * `  ( A  .ih  B ) )  x.  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B ) )
271, 3his1i 22555 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
2827eqcomi 2408 . . . . . . . . . . 11  |-  ( * `
 ( A  .ih  B ) )  =  ( B  .ih  A )
29 ax-his3 22539 . . . . . . . . . . . 12  |-  ( ( ( B  .ih  B
)  e.  CC  /\  A  e.  ~H  /\  B  e.  ~H )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) ) )
302, 3, 1, 29mp3an 1279 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
3128, 30oveq12i 6052 . . . . . . . . . 10  |-  ( ( * `  ( A 
.ih  B ) )  x.  ( ( ( B  .ih  B )  .h  A )  .ih  B ) )  =  ( ( B  .ih  A
)  x.  ( ( B  .ih  B )  x.  ( A  .ih  B ) ) )
321, 3hicli 22536 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  e.  CC
332, 5mulcli 9051 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  e.  CC
3432, 33mulcomi 9052 . . . . . . . . . . 11  |-  ( ( B  .ih  A )  x.  ( ( B 
.ih  B )  x.  ( A  .ih  B
) ) )  =  ( ( ( B 
.ih  B )  x.  ( A  .ih  B
) )  x.  ( B  .ih  A ) )
352, 5, 32mulassi 9055 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  x.  ( A 
.ih  B ) )  x.  ( B  .ih  A ) )  =  ( ( B  .ih  B
)  x.  ( ( A  .ih  B )  x.  ( B  .ih  A ) ) )
365, 32mulcli 9051 . . . . . . . . . . . 12  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
372, 36mulcomi 9052 . . . . . . . . . . 11  |-  ( ( B  .ih  B )  x.  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )  =  ( ( ( A 
.ih  B )  x.  ( B  .ih  A
) )  x.  ( B  .ih  B ) )
3834, 35, 373eqtri 2428 . . . . . . . . . 10  |-  ( ( B  .ih  A )  x.  ( ( B 
.ih  B )  x.  ( A  .ih  B
) ) )  =  ( ( ( A 
.ih  B )  x.  ( B  .ih  A
) )  x.  ( B  .ih  B ) )
3926, 31, 383eqtri 2428 . . . . . . . . 9  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( ( A 
.ih  B )  x.  ( B  .ih  A
) )  x.  ( B  .ih  B ) )
409, 24, 393eqtr4g 2461 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( ( B 
.ih  B )  .h  A )  .ih  (
( A  .ih  B
)  .h  B ) ) )
41 ax-his3 22539 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  e.  CC  /\  B  e.  ~H  /\  A  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) ) )
425, 1, 3, 41mp3an 1279 . . . . . . . . . . 11  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
4314, 42oveq12i 6052 . . . . . . . . . 10  |-  ( ( * `  ( B 
.ih  B ) )  x.  ( ( ( A  .ih  B )  .h  B )  .ih  A ) )  =  ( ( B  .ih  B
)  x.  ( ( A  .ih  B )  x.  ( B  .ih  A ) ) )
44 his5 22541 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  e.  CC  /\  ( ( A  .ih  B )  .h  B )  e.  ~H  /\  A  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( * `  ( B  .ih  B ) )  x.  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A ) ) )
452, 6, 3, 44mp3an 1279 . . . . . . . . . 10  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  =  ( ( * `  ( B  .ih  B ) )  x.  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A ) )
46 his5 22541 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  e.  CC  /\  ( ( A  .ih  B )  .h  B )  e.  ~H  /\  B  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( * `  ( A  .ih  B ) )  x.  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B ) ) )
475, 6, 1, 46mp3an 1279 . . . . . . . . . . 11  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( * `  ( A  .ih  B ) )  x.  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B ) )
48 ax-his3 22539 . . . . . . . . . . . . 13  |-  ( ( ( A  .ih  B
)  e.  CC  /\  B  e.  ~H  /\  B  e.  ~H )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) ) )
495, 1, 1, 48mp3an 1279 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
5028, 49oveq12i 6052 . . . . . . . . . . 11  |-  ( ( * `  ( A 
.ih  B ) )  x.  ( ( ( A  .ih  B )  .h  B )  .ih  B ) )  =  ( ( B  .ih  A
)  x.  ( ( A  .ih  B )  x.  ( B  .ih  B ) ) )
515, 2mulcli 9051 . . . . . . . . . . . . 13  |-  ( ( A  .ih  B )  x.  ( B  .ih  B ) )  e.  CC
5232, 51mulcomi 9052 . . . . . . . . . . . 12  |-  ( ( B  .ih  A )  x.  ( ( A 
.ih  B )  x.  ( B  .ih  B
) ) )  =  ( ( ( A 
.ih  B )  x.  ( B  .ih  B
) )  x.  ( B  .ih  A ) )
535, 2, 32mul32i 9218 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  B ) )  x.  ( B  .ih  A ) )  =  ( ( ( A  .ih  B )  x.  ( B 
.ih  A ) )  x.  ( B  .ih  B ) )
5436, 2mulcomi 9052 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  x.  ( ( A  .ih  B )  x.  ( B  .ih  A ) ) )
5552, 53, 543eqtri 2428 . . . . . . . . . . 11  |-  ( ( B  .ih  A )  x.  ( ( A 
.ih  B )  x.  ( B  .ih  B
) ) )  =  ( ( B  .ih  B )  x.  ( ( A  .ih  B )  x.  ( B  .ih  A ) ) )
5647, 50, 553eqtri 2428 . . . . . . . . . 10  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( B  .ih  B )  x.  ( ( A  .ih  B )  x.  ( B  .ih  A ) ) )
5743, 45, 563eqtr4ri 2435 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) )
5857a1i 11 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( A  .ih  B )  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) )
5940, 58oveq12d 6058 . . . . . . 7  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( ( B 
.ih  B )  .h  A )  .ih  (
( B  .ih  B
)  .h  A ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) ) )  =  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) ) )
6059oveq1d 6055 . . . . . 6  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( ( ( B  .ih  B )  .h  A )  .ih  ( ( B  .ih  B )  .h  A ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) ) )  -  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) ) )  =  ( ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) )  -  (
( ( ( B 
.ih  B )  .h  A )  .ih  (
( A  .ih  B
)  .h  B ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) ) ) ) )
614, 6hicli 22536 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  e.  CC
626, 4hicli 22536 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  e.  CC
6361, 62addcli 9050 . . . . . . 7  |-  ( ( ( ( B  .ih  B )  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) )  e.  CC
6463subidi 9327 . . . . . 6  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  .ih  (
( A  .ih  B
)  .h  B ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) ) )  -  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) ) )  =  0
6560, 64syl6eq 2452 . . . . 5  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( ( ( B  .ih  B )  .h  A )  .ih  ( ( B  .ih  B )  .h  A ) )  +  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) ) )  -  ( ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  +  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) ) ) )  =  0 )
667, 65syl5eq 2448 . . . 4  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) ) )  =  0 )
674, 6hvsubcli 22477 . . . . 5  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
68 his6 22554 . . . . 5  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H  ->  ( ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  .ih  ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) ) )  =  0  <-> 
( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  =  0h )
)
6967, 68ax-mp 8 . . . 4  |-  ( ( ( ( ( B 
.ih  B )  .h  A )  -h  (
( A  .ih  B
)  .h  B ) )  .ih  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) ) )  =  0  <->  ( (
( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  =  0h )
7066, 69sylib 189 . . 3  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  =  0h )
714, 6hvsubeq0i 22518 . . 3  |-  ( ( ( ( B  .ih  B )  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  =  0h  <->  ( ( B 
.ih  B )  .h  A )  =  ( ( A  .ih  B
)  .h  B ) )
7270, 71sylib 189 . 2  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  ->  (
( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B ) )
73 oveq1 6047 . . . 4  |-  ( ( ( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B )  ->  (
( ( B  .ih  B )  .h  A ) 
.ih  A )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  A
) )
7421, 16eqtr4i 2427 . . . 4  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( ( B  .ih  B )  .h  A ) 
.ih  A )
7542eqcomi 2408 . . . 4  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  =  ( ( ( A  .ih  B )  .h  B ) 
.ih  A )
7673, 74, 753eqtr4g 2461 . . 3  |-  ( ( ( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B )  ->  (
( A  .ih  A
)  x.  ( B 
.ih  B ) )  =  ( ( A 
.ih  B )  x.  ( B  .ih  A
) ) )
7776eqcomd 2409 . 2  |-  ( ( ( B  .ih  B
)  .h  A )  =  ( ( A 
.ih  B )  .h  B )  ->  (
( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) ) )
7872, 77impbii 181 1  |-  ( ( ( A  .ih  B
)  x.  ( B 
.ih  A ) )  =  ( ( A 
.ih  A )  x.  ( B  .ih  B
) )  <->  ( ( B  .ih  B )  .h  A )  =  ( ( A  .ih  B
)  .h  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    + caddc 8949    x. cmul 8951    - cmin 9247   *ccj 11856   ~Hchil 22375    .h csm 22377    .ih csp 22378   0hc0v 22380    -h cmv 22381
This theorem is referenced by:  h1de2i  23008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861  df-hvsub 22427
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