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Theorem dipcj 21404
Description: The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipcl.1  |-  X  =  ( BaseSet `  U )
ipcl.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipcj  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )

Proof of Theorem dipcj
StepHypRef Expression
1 ipcl.1 . . . 4  |-  X  =  ( BaseSet `  U )
2 eqid 2358 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2358 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2358 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 21394 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )
76fveq2d 5612 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( * `  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) ) )
81, 2, 3, 4, 5ipval2 21394 . . . 4  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  ( B P A )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
983com23 1157 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( B P A )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
101, 2, 3, 4, 5ipval2lem3 21392 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  RR )
1110recnd 8951 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  CC )
12 neg1cn 9903 . . . . . . . 8  |-  -u 1  e.  CC
131, 2, 3, 4, 5ipval2lem4 21393 . . . . . . . 8  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u 1  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 )  e.  CC )
1412, 13mpan2 652 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1511, 14subcld 9247 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC )
16 ax-icn 8886 . . . . . . 7  |-  _i  e.  CC
171, 2, 3, 4, 5ipval2lem4 21393 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1816, 17mpan2 652 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1916negcli 9204 . . . . . . . . 9  |-  -u _i  e.  CC
201, 2, 3, 4, 5ipval2lem4 21393 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
2119, 20mpan2 652 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
2218, 21subcld 9247 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )
23 mulcl 8911 . . . . . . 7  |-  ( ( _i  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( _i  x.  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  e.  CC )
2416, 22, 23sylancr 644 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  e.  CC )
2515, 24addcld 8944 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC )
26 4cn 9910 . . . . . 6  |-  4  e.  CC
27 4re 9909 . . . . . . 7  |-  4  e.  RR
28 4pos 9922 . . . . . . 7  |-  0  <  4
2927, 28gt0ne0ii 9399 . . . . . 6  |-  4  =/=  0
30 cjdiv 11745 . . . . . 6  |-  ( ( ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC  /\  4  e.  CC  /\  4  =/=  0 )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
3126, 29, 30mp3an23 1269 . . . . 5  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  e.  CC  ->  ( * `  ( ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
3225, 31syl 15 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) ) )
33 cjre 11720 . . . . . . 7  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
3427, 33ax-mp 8 . . . . . 6  |-  ( * `
 4 )  =  4
3534oveq2i 5956 . . . . 5  |-  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) )  =  ( ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  4 )
361, 2, 3, 4, 5ipval2lem2 21391 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u 1  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 )  e.  RR )
3712, 36mpan2 652 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
3810, 37resubcld 9301 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  RR )
391, 2, 3, 4, 5ipval2lem2 21391 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4016, 39mpan2 652 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
411, 2, 3, 4, 5ipval2lem2 21391 . . . . . . . . . 10  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  -u _i  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4219, 41mpan2 652 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  RR )
4340, 42resubcld 9301 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  RR )
44 cjreim 11741 . . . . . . . 8  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  RR  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  RR )  ->  ( * `  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4538, 43, 44syl2anc 642 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
46 submul2 9310 . . . . . . . . 9  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC  /\  _i  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4716, 46mp3an2 1265 . . . . . . . 8  |-  ( ( ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  e.  CC  /\  ( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )  ->  ( ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
4815, 22, 47syl2anc 642 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  -  (
_i  x.  ( (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )
491, 2nvcom 21291 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) B )  =  ( B ( +v `  U ) A ) )
5049fveq2d 5612 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) B ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) A ) ) )
5150oveq1d 5960 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  =  ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 ) )
521, 2, 3, 4nvdif 21345 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) )  =  ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) )
5352oveq1d 5960 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u 1
( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )
5451, 53oveq12d 5963 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) ) )
5518, 21negsubdi2d 9263 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )
561, 2, 3, 4nvpi 21346 . . . . . . . . . . . . . 14  |-  ( ( U  e.  NrmCVec  /\  B  e.  X  /\  A  e.  X )  ->  (
( normCV `  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) )  =  ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) )
57563com23 1157 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) )  =  ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) )
5857eqcomd 2363 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) )
5958oveq1d 5960 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )
601, 2, 3, 4nvpi 21346 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) )
6160oveq1d 5960 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  =  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) )
6259, 61oveq12d 5963 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )
6355, 62eqtrd 2390 . . . . . . . . 9  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) )  =  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) )
6463oveq2d 5961 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
_i  x.  -u ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) )  =  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )
6554, 64oveq12d 5963 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  -u (
( ( ( normCV `  U ) `  ( A ( +v `  U ) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  =  ( ( ( ( ( normCV `  U ) `  ( B ( +v `  U ) A ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) ) )
6645, 48, 653eqtrd 2394 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  =  ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) ) )
6766oveq1d 5960 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( * `  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  4 )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
6835, 67syl5eq 2402 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( * `  (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) ) )  /  ( * ` 
4 ) )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
6932, 68eqtrd 2390 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( (
( ( ( (
normCV
`  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) )  =  ( ( ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) A ) ) ^ 2 )  -  ( ( ( normCV `  U ) `  ( B ( +v `  U ) ( -u
1 ( .s OLD `  U ) A ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( B
( +v `  U
) ( _i ( .s OLD `  U
) A ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( B ( +v `  U ) ( -u _i ( .s OLD `  U
) A ) ) ) ^ 2 ) ) ) )  / 
4 ) )
709, 69eqtr4d 2393 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( B P A )  =  ( * `  (
( ( ( ( ( normCV `  U ) `  ( A ( +v `  U ) B ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u
1 ( .s OLD `  U ) B ) ) ) ^ 2 ) )  +  ( _i  x.  ( ( ( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  -  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( -u _i ( .s OLD `  U
) B ) ) ) ^ 2 ) ) ) )  / 
4 ) ) )
717, 70eqtr4d 2393 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
* `  ( A P B ) )  =  ( B P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828   _ici 8829    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128    / cdiv 9513   2c2 9885   4c4 9887   ^cexp 11197   *ccj 11677   NrmCVeccnv 21254   +vcpv 21255   BaseSetcba 21256   .s
OLDcns 21257   normCVcnmcv 21260   .i OLDcdip 21387
This theorem is referenced by:  ipipcj  21405  diporthcom  21406  dip0l  21408  ipasslem10  21531  dipdi  21535  dipassr  21538  dipsubdi  21541  siii  21545  hlipcj  21604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-grpo 20970  df-gid 20971  df-ginv 20972  df-ablo 21061  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-nmcv 21270  df-dip 21388
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