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Theorem bcseqi 21699
Description: Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 21759. (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 21660 . . . . . . 7  |-  ( B 
.ih  B )  e.  CC
3 normlem7t.1 . . . . . . 7  |-  A  e. 
~H
42, 3hvmulcli 21594 . . . . . 6  |-  ( ( B  .ih  B )  .h  A )  e. 
~H
53, 1hicli 21660 . . . . . . 7  |-  ( A 
.ih  B )  e.  CC
65, 1hvmulcli 21594 . . . . . 6  |-  ( ( A  .ih  B )  .h  B )  e. 
~H
74, 6, 4, 6normlem9 21697 . . . . 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 5865 . . . . . . . . . 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 2288 . . . . . . . . 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 21665 . . . . . . . . . . 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 1277 . . . . . . . . . 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 21674 . . . . . . . . . . . 12  |-  ( B  e.  ~H  ->  ( B  .ih  B )  e.  RR )
13 cjre 11624 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  e.  RR  ->  (
* `  ( B  .ih  B ) )  =  ( B  .ih  B
) )
141, 12, 13mp2b 9 . . . . . . . . . . 11  |-  ( * `
 ( B  .ih  B ) )  =  ( B  .ih  B )
15 ax-his3 21663 . . . . . . . . . . . 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 1277 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  A )  =  ( ( B  .ih  B )  x.  ( A 
.ih  A ) )
1714, 16oveq12i 5870 . . . . . . . . . 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 21660 . . . . . . . . . . . . 13  |-  ( A 
.ih  A )  e.  CC
192, 18mulcli 8842 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  A ) )  e.  CC
202, 19mulcomi 8843 . . . . . . . . . . 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 8843 . . . . . . . . . . . 12  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( B  .ih  B
)  x.  ( A 
.ih  A ) )
2221oveq1i 5868 . . . . . . . . . . 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 2306 . . . . . . . . . 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 2307 . . . . . . . . 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 21665 . . . . . . . . . . 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 1277 . . . . . . . . . 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 21679 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  =  ( * `  ( A  .ih  B ) )
2827eqcomi 2287 . . . . . . . . . . 11  |-  ( * `
 ( A  .ih  B ) )  =  ( B  .ih  A )
29 ax-his3 21663 . . . . . . . . . . . 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 1277 . . . . . . . . . . 11  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  B )  =  ( ( B  .ih  B )  x.  ( A 
.ih  B ) )
3128, 30oveq12i 5870 . . . . . . . . . 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 21660 . . . . . . . . . . . 12  |-  ( B 
.ih  A )  e.  CC
332, 5mulcli 8842 . . . . . . . . . . . 12  |-  ( ( B  .ih  B )  x.  ( A  .ih  B ) )  e.  CC
3432, 33mulcomi 8843 . . . . . . . . . . 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 8846 . . . . . . . . . . 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 8842 . . . . . . . . . . . 12  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  e.  CC
372, 36mulcomi 8843 . . . . . . . . . . 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 2307 . . . . . . . . . 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 2307 . . . . . . . . 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 2340 . . . . . . . 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 21663 . . . . . . . . . . . 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 1277 . . . . . . . . . . 11  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  A )  =  ( ( A  .ih  B )  x.  ( B 
.ih  A ) )
4314, 42oveq12i 5870 . . . . . . . . . 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 21665 . . . . . . . . . . 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 1277 . . . . . . . . . 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 21665 . . . . . . . . . . . 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 1277 . . . . . . . . . . 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 21663 . . . . . . . . . . . . 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 1277 . . . . . . . . . . . 12  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  B )  =  ( ( A  .ih  B )  x.  ( B 
.ih  B ) )
5028, 49oveq12i 5870 . . . . . . . . . . 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 8842 . . . . . . . . . . . . 13  |-  ( ( A  .ih  B )  x.  ( B  .ih  B ) )  e.  CC
5232, 51mulcomi 8843 . . . . . . . . . . . 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 9008 . . . . . . . . . . . 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 8843 . . . . . . . . . . . 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 2307 . . . . . . . . . . 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 2307 . . . . . . . . . 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 2314 . . . . . . . . 9  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  =  ( ( ( A 
.ih  B )  .h  B )  .ih  (
( B  .ih  B
)  .h  A ) )
5857a1i 10 . . . . . . . 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 5876 . . . . . . 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 5873 . . . . . 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 21660 . . . . . . . 8  |-  ( ( ( B  .ih  B
)  .h  A ) 
.ih  ( ( A 
.ih  B )  .h  B ) )  e.  CC
626, 4hicli 21660 . . . . . . . 8  |-  ( ( ( A  .ih  B
)  .h  B ) 
.ih  ( ( B 
.ih  B )  .h  A ) )  e.  CC
6361, 62addcli 8841 . . . . . . 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 9117 . . . . . 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 2331 . . . . 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 2327 . . . 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 21601 . . . . 5  |-  ( ( ( B  .ih  B
)  .h  A )  -h  ( ( A 
.ih  B )  .h  B ) )  e. 
~H
68 his6 21678 . . . . 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 188 . . 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 21642 . . 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 188 . 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 5865 . . . 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 2306 . . . 4  |-  ( ( A  .ih  A )  x.  ( B  .ih  B ) )  =  ( ( ( B  .ih  B )  .h  A ) 
.ih  A )
7542eqcomi 2287 . . . 4  |-  ( ( A  .ih  B )  x.  ( B  .ih  A ) )  =  ( ( ( A  .ih  B )  .h  B ) 
.ih  A )
7673, 74, 753eqtr4g 2340 . . 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 2288 . 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 180 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 176    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    + caddc 8740    x. cmul 8742    - cmin 9037   *ccj 11581   ~Hchil 21499    .h csm 21501    .ih csp 21502   0hc0v 21504    -h cmv 21505
This theorem is referenced by:  h1de2i  22132
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551
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