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Theorem dipcj 22218
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 2438 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2438 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2438 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 22208 . . 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 5735 . 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 22208 . . . 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 1160 . . 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 22206 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  RR )
1110recnd 9119 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  CC )
12 neg1cn 10072 . . . . . . . 8  |-  -u 1  e.  CC
131, 2, 3, 4, 5ipval2lem4 22207 . . . . . . . 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 654 . . . . . . 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 9416 . . . . . 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 9054 . . . . . . 7  |-  _i  e.  CC
171, 2, 3, 4, 5ipval2lem4 22207 . . . . . . . . 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 654 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1916negcli 9373 . . . . . . . . 9  |-  -u _i  e.  CC
201, 2, 3, 4, 5ipval2lem4 22207 . . . . . . . . 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 654 . . . . . . . 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 9416 . . . . . . 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 9079 . . . . . . 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 646 . . . . . 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 9112 . . . . 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 10079 . . . . . 6  |-  4  e.  CC
27 4re 10078 . . . . . . 7  |-  4  e.  RR
28 4pos 10091 . . . . . . 7  |-  0  <  4
2927, 28gt0ne0ii 9568 . . . . . 6  |-  4  =/=  0
30 cjdiv 11974 . . . . . 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 1272 . . . . 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 11949 . . . . . . 7  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
3427, 33ax-mp 5 . . . . . 6  |-  ( * `
 4 )  =  4
3534oveq2i 6095 . . . . 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 22205 . . . . . . . . . 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 654 . . . . . . . . 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 9470 . . . . . . . 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 22205 . . . . . . . . . 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 654 . . . . . . . . 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 22205 . . . . . . . . . 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 654 . . . . . . . . 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 9470 . . . . . . . 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 11970 . . . . . . . 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 644 . . . . . . 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 9479 . . . . . . . . 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 1268 . . . . . . . 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 644 . . . . . . 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 22105 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) B )  =  ( B ( +v `  U ) A ) )
5049fveq2d 5735 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) B ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) A ) ) )
5150oveq1d 6099 . . . . . . . . 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 22159 . . . . . . . . . 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 6099 . . . . . . . . 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 6102 . . . . . . . 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 9432 . . . . . . . . . 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 22160 . . . . . . . . . . . . . 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 1160 . . . . . . . . . . . . 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 2443 . . . . . . . . . . . 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 6099 . . . . . . . . . . 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 22160 . . . . . . . . . . . 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 6099 . . . . . . . . . . 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 6102 . . . . . . . . . 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 2470 . . . . . . . . 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 6100 . . . . . . . 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 6102 . . . . . . 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 2474 . . . . . 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 6099 . . . . 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 2482 . . . 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 2470 . . 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 2473 . 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 2473 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 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996   _ici 8997    + caddc 8998    x. cmul 9000    - cmin 9296   -ucneg 9297    / cdiv 9682   2c2 10054   4c4 10056   ^cexp 11387   *ccj 11906   NrmCVeccnv 22068   +vcpv 22069   BaseSetcba 22070   .s
OLDcns 22071   normCVcnmcv 22074   .i OLDcdip 22201
This theorem is referenced by:  ipipcj  22219  diporthcom  22220  dip0l  22222  ipasslem10  22345  dipdi  22349  dipassr  22352  dipsubdi  22355  siii  22359  hlipcj  22418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-vc 22030  df-nv 22076  df-va 22079  df-ba 22080  df-sm 22081  df-0v 22082  df-nmcv 22084  df-dip 22202
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