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Theorem bcseqi 22463
Description: Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 22523. (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 22424 . . . . . . 7  |-  ( B 
.ih  B )  e.  CC
3 normlem7t.1 . . . . . . 7  |-  A  e. 
~H
42, 3hvmulcli 22358 . . . . . 6  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 22424 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 22358 . . . . . 6  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
74, 6, 4, 6normlem9 22461 . . . . 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 6020 . . . . . . . . . 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 2385 . . . . . . . . 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 22429 . . . . . . . . . . 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 22438 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
13 cjre 11864 . . . . . . . . . . . 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 22427 . . . . . . . . . . . 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 6025 . . . . . . . . . 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 22424 . . . . . . . . . . . . 13  |-  ( A 
.ih  A )  e.  CC
192, 18mulcli 9021 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  e.  CC
202, 19mulcomi 9022 . . . . . . . . . . 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 9022 . . . . . . . . . . . 12  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  x.  ( A 
.ih  A ) )
2221oveq1i 6023 . . . . . . . . . . 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 2403 . . . . . . . . . 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 2404 . . . . . . . . 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 22429 . . . . . . . . . . 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 22443 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
2827eqcomi 2384 . . . . . . . . . . 11  |-  ( * `
 ( A  .ih  B ) )  =  ( B  .ih  A )
29 ax-his3 22427 . . . . . . . . . . . 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 6025 . . . . . . . . . 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 22424 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  e.  CC
332, 5mulcli 9021 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  e.  CC
3432, 33mulcomi 9022 . . . . . . . . . . 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 9025 . . . . . . . . . . 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 9021 . . . . . . . . . . . 12  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
372, 36mulcomi 9022 . . . . . . . . . . 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 2404 . . . . . . . . . 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 2404 . . . . . . . . 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 2437 . . . . . . . 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 22427 . . . . . . . . . . . 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 6025 . . . . . . . . . 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 22429 . . . . . . . . . . 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 22429 . . . . . . . . . . . 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 22427 . . . . . . . . . . . . 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 6025 . . . . . . . . . . 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 9021 . . . . . . . . . . . . 13  |-  ( ( A  .ih  B )  x.  ( B  .ih  B ) )  e.  CC
5232, 51mulcomi 9022 . . . . . . . . . . . 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 9187 . . . . . . . . . . . 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 9022 . . . . . . . . . . . 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 2404 . . . . . . . . . . 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 2404 . . . . . . . . . 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 2411 . . . . . . . . 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 6031 . . . . . . 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 6028 . . . . . 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 22424 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  e.  CC
626, 4hicli 22424 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  e.  CC
6361, 62addcli 9020 . . . . . . 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 9296 . . . . . 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 2428 . . . . 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 2424 . . . 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 22365 . . . . 5  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
68 his6 22442 . . . . 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 22406 . . 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 6020 . . . 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 2403 . . . 4  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( ( B  .ih  B )  .h  A ) 
.ih  A )
7542eqcomi 2384 . . . 4  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  =  ( ( ( A  .ih  B )  .h  B ) 
.ih  A )
7673, 74, 753eqtr4g 2437 . . 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 2385 . 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 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916    + caddc 8919    x. cmul 8921    - cmin 9216   *ccj 11821   ~Hchil 22263    .h csm 22265    .ih csp 22266   0hc0v 22268    -h cmv 22269
This theorem is referenced by:  h1de2i  22896
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-hfvadd 22344  ax-hvcom 22345  ax-hvass 22346  ax-hv0cl 22347  ax-hvaddid 22348  ax-hfvmul 22349  ax-hvmulid 22350  ax-hvdistr2 22353  ax-hvmul0 22354  ax-hfi 22422  ax-his1 22425  ax-his2 22426  ax-his3 22427  ax-his4 22428
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983  df-cj 11824  df-re 11825  df-im 11826  df-hvsub 22315
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