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Theorem dipcj 21120
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 2253 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2253 . . . 4  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2253 . . . 4  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . 4  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval2 21110 . . 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 5381 . 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 21110 . . . 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 1162 . . 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 21108 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  RR )
1110recnd 8741 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) B ) ) ^ 2 )  e.  CC )
12 neg1cn 9693 . . . . . . . 8  |-  -u 1  e.  CC
131, 2, 3, 4, 5ipval2lem4 21109 . . . . . . . 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 655 . . . . . . 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 9037 . . . . . 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 8676 . . . . . . 7  |-  _i  e.  CC
171, 2, 3, 4, 5ipval2lem4 21109 . . . . . . . . 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 655 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( _i ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1916negcli 8994 . . . . . . . . 9  |-  -u _i  e.  CC
201, 2, 3, 4, 5ipval2lem4 21109 . . . . . . . . 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 655 . . . . . . . 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 9037 . . . . . . 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 8701 . . . . . . 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 647 . . . . . 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 8734 . . . . 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 9700 . . . . . 6  |-  4  e.  CC
27 4re 9699 . . . . . . 7  |-  4  e.  RR
28 4pos 9712 . . . . . . 7  |-  0  <  4
2927, 28gt0ne0ii 9189 . . . . . 6  |-  4  =/=  0
30 cjdiv 11526 . . . . . 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 1274 . . . . 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 17 . . . 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 11501 . . . . . . 7  |-  ( 4  e.  RR  ->  (
* `  4 )  =  4 )
3427, 33ax-mp 10 . . . . . 6  |-  ( * `
 4 )  =  4
3534oveq2i 5721 . . . . 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 21107 . . . . . . . . . 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 655 . . . . . . . . 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 9091 . . . . . . . 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 21107 . . . . . . . . . 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 655 . . . . . . . . 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 21107 . . . . . . . . . 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 655 . . . . . . . . 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 9091 . . . . . . . 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 11522 . . . . . . . 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 645 . . . . . . 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 9100 . . . . . . . . 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 1270 . . . . . . . 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 645 . . . . . . 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 21007 . . . . . . . . . . 11  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A ( +v `  U ) B )  =  ( B ( +v `  U ) A ) )
5049fveq2d 5381 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( normCV `  U ) `  ( A ( +v `  U ) B ) )  =  ( (
normCV
`  U ) `  ( B ( +v `  U ) A ) ) )
5150oveq1d 5725 . . . . . . . . 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 21061 . . . . . . . . . 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 5725 . . . . . . . . 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 5728 . . . . . . . 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 9053 . . . . . . . . . 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 21062 . . . . . . . . . . . . . 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 1162 . . . . . . . . . . . . 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 2258 . . . . . . . . . . . 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 5725 . . . . . . . . . . 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 21062 . . . . . . . . . . . 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 5725 . . . . . . . . . . 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 5728 . . . . . . . . . 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 2285 . . . . . . . . 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 5726 . . . . . . . 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 5728 . . . . . . 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 2289 . . . . . 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 5725 . . . . 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 2297 . . . 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 2285 . . 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 2288 . 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 2288 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 6    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618   _ici 8619    + caddc 8620    x. cmul 8622    - cmin 8917   -ucneg 8918    / cdiv 9303   2c2 9675   4c4 9677   ^cexp 10982   *ccj 11458   NrmCVeccnv 20970   +vcpv 20971   BaseSetcba 20972   .s
OLDcns 20973   normCVcnmcv 20976   .i OLDcdip 21103
This theorem is referenced by:  ipipcj  21121  diporthcom  21122  dip0l  21124  ipasslem10  21247  dipdi  21251  dipassr  21254  dipsubdi  21257  siii  21261  hlipcj  21320
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-grpo 20688  df-gid 20689  df-ginv 20690  df-ablo 20779  df-vc 20932  df-nv 20978  df-va 20981  df-ba 20982  df-sm 20983  df-0v 20984  df-nmcv 20986  df-dip 21104
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