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Theorem dipcj 22170
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 2408 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2408 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2408 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 22160 . . 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 5695 . 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 22160 . . . 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 1159 . . 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 22158 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  RR )
1110recnd 9074 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  CC )
12 neg1cn 10027 . . . . . . . 8  |-  -u 1  e.  CC
131, 2, 3, 4, 5ipval2lem4 22159 . . . . . . . 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 653 . . . . . . 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 9371 . . . . . 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 9009 . . . . . . 7  |-  _i  e.  CC
171, 2, 3, 4, 5ipval2lem4 22159 . . . . . . . . 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 653 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1916negcli 9328 . . . . . . . . 9  |-  -u _i  e.  CC
201, 2, 3, 4, 5ipval2lem4 22159 . . . . . . . . 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 653 . . . . . . . 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 9371 . . . . . . 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 9034 . . . . . . 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 645 . . . . . 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 9067 . . . . 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 10034 . . . . . 6  |-  4  e.  CC
27 4re 10033 . . . . . . 7  |-  4  e.  RR
28 4pos 10046 . . . . . . 7  |-  0  <  4
2927, 28gt0ne0ii 9523 . . . . . 6  |-  4  =/=  0
30 cjdiv 11928 . . . . . 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 1271 . . . . 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 16 . . . 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 11903 . . . . . . 7  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
3427, 33ax-mp 8 . . . . . 6  |-  ( * `
 4 )  =  4
3534oveq2i 6055 . . . . 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 22157 . . . . . . . . . 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 653 . . . . . . . . 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 9425 . . . . . . . 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 22157 . . . . . . . . . 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 653 . . . . . . . . 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 22157 . . . . . . . . . 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 653 . . . . . . . . 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 9425 . . . . . . . 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 11924 . . . . . . . 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 643 . . . . . . 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 9434 . . . . . . . . 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 1267 . . . . . . . 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 643 . . . . . . 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 22057 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) B )  =  ( B ( +v `  U ) A ) )
5049fveq2d 5695 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) B ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) A ) ) )
5150oveq1d 6059 . . . . . . . . 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 22111 . . . . . . . . . 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 6059 . . . . . . . . 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 6062 . . . . . . . 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 9387 . . . . . . . . . 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 22112 . . . . . . . . . . . . . 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 1159 . . . . . . . . . . . . 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 2413 . . . . . . . . . . . 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 6059 . . . . . . . . . . 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 22112 . . . . . . . . . . . 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 6059 . . . . . . . . . . 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 6062 . . . . . . . . . 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 2440 . . . . . . . . 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 6060 . . . . . . . 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 6062 . . . . . . 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 2444 . . . . . 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 6059 . . . . 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 2452 . . . 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 2440 . . 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 2443 . 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 2443 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 936    = wceq 1649    e. wcel 1721    =/= wne 2571   ` cfv 5417  (class class class)co 6044   CCcc 8948   RRcr 8949   0cc0 8950   1c1 8951   _ici 8952    + caddc 8953    x. cmul 8955    - cmin 9251   -ucneg 9252    / cdiv 9637   2c2 10009   4c4 10011   ^cexp 11341   *ccj 11860   NrmCVeccnv 22020   +vcpv 22021   BaseSetcba 22022   .s
OLDcns 22023   normCVcnmcv 22026   .i OLDcdip 22153
This theorem is referenced by:  ipipcj  22171  diporthcom  22172  dip0l  22174  ipasslem10  22297  dipdi  22301  dipassr  22304  dipsubdi  22307  siii  22311  hlipcj  22370
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-nmcv 22036  df-dip 22154
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