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Theorem bcseqi 22614
Description: Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 22674. (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 22575 . . . . . . 7  |-  ( B 
.ih  B )  e.  CC
3 normlem7t.1 . . . . . . 7  |-  A  e. 
~H
42, 3hvmulcli 22509 . . . . . 6  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 22575 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 22509 . . . . . 6  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
74, 6, 4, 6normlem9 22612 . . . . 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 6080 . . . . . . . . . 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 2440 . . . . . . . . 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 22580 . . . . . . . . . . 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 22589 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
13 cjre 11936 . . . . . . . . . . . 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 22578 . . . . . . . . . . . 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 6085 . . . . . . . . . 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 22575 . . . . . . . . . . . . 13  |-  ( A 
.ih  A )  e.  CC
192, 18mulcli 9087 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  e.  CC
202, 19mulcomi 9088 . . . . . . . . . . 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 9088 . . . . . . . . . . . 12  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  x.  ( A 
.ih  A ) )
2221oveq1i 6083 . . . . . . . . . . 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 2458 . . . . . . . . . 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 2459 . . . . . . . . 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 22580 . . . . . . . . . . 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 22594 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
2827eqcomi 2439 . . . . . . . . . . 11  |-  ( * `
 ( A  .ih  B ) )  =  ( B  .ih  A )
29 ax-his3 22578 . . . . . . . . . . . 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 6085 . . . . . . . . . 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 22575 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  e.  CC
332, 5mulcli 9087 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  e.  CC
3432, 33mulcomi 9088 . . . . . . . . . . 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 9091 . . . . . . . . . . 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 9087 . . . . . . . . . . . 12  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
372, 36mulcomi 9088 . . . . . . . . . . 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 2459 . . . . . . . . . 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 2459 . . . . . . . . 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 2492 . . . . . . . 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 22578 . . . . . . . . . . . 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 6085 . . . . . . . . . 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 22580 . . . . . . . . . . 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 22580 . . . . . . . . . . . 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 22578 . . . . . . . . . . . . 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 6085 . . . . . . . . . . 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 9087 . . . . . . . . . . . . 13  |-  ( ( A  .ih  B )  x.  ( B  .ih  B ) )  e.  CC
5232, 51mulcomi 9088 . . . . . . . . . . . 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 9254 . . . . . . . . . . . 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 9088 . . . . . . . . . . . 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 2459 . . . . . . . . . . 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 2459 . . . . . . . . . 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 2466 . . . . . . . . 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 6091 . . . . . . 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 6088 . . . . . 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 22575 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  e.  CC
626, 4hicli 22575 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  e.  CC
6361, 62addcli 9086 . . . . . . 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 9363 . . . . . 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 2483 . . . . 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 2479 . . . 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 22516 . . . . 5  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
68 his6 22593 . . . . 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 22557 . . 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 6080 . . . 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 2458 . . . 4  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( ( B  .ih  B )  .h  A ) 
.ih  A )
7542eqcomi 2439 . . . 4  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  =  ( ( ( A  .ih  B )  .h  B ) 
.ih  A )
7673, 74, 753eqtr4g 2492 . . 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 2440 . 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    + caddc 8985    x. cmul 8987    - cmin 9283   *ccj 11893   ~Hchil 22414    .h csm 22416    .ih csp 22417   0hc0v 22419    -h cmv 22420
This theorem is referenced by:  h1de2i  23047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-hfvadd 22495  ax-hvcom 22496  ax-hvass 22497  ax-hv0cl 22498  ax-hvaddid 22499  ax-hfvmul 22500  ax-hvmulid 22501  ax-hvdistr2 22504  ax-hvmul0 22505  ax-hfi 22573  ax-his1 22576  ax-his2 22577  ax-his3 22578  ax-his4 22579
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-2 10050  df-cj 11896  df-re 11897  df-im 11898  df-hvsub 22466
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