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Theorem bj-diagval 31431
Description: Value of the diagonal. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-diagval  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )

Proof of Theorem bj-diagval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 3087 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 incom 3652 . . 3  |-  ( ( A  X.  A )  i^i  _I  )  =  (  _I  i^i  ( A  X.  A ) )
3 sqxpexg 6601 . . . 4  |-  ( A  e.  V  ->  ( A  X.  A )  e. 
_V )
4 inex1g 4559 . . . 4  |-  ( ( A  X.  A )  e.  _V  ->  (
( A  X.  A
)  i^i  _I  )  e.  _V )
53, 4syl 17 . . 3  |-  ( A  e.  V  ->  (
( A  X.  A
)  i^i  _I  )  e.  _V )
62, 5syl5eqelr 2513 . 2  |-  ( A  e.  V  ->  (  _I  i^i  ( A  X.  A ) )  e. 
_V )
7 id 23 . . . . 5  |-  ( x  =  A  ->  x  =  A )
87sqxpeqd 4871 . . . 4  |-  ( x  =  A  ->  (
x  X.  x )  =  ( A  X.  A ) )
98ineq2d 3661 . . 3  |-  ( x  =  A  ->  (  _I  i^i  ( x  X.  x ) )  =  (  _I  i^i  ( A  X.  A ) ) )
10 df-bj-diag 31430 . . 3  |- Diag  =  ( x  e.  _V  |->  (  _I  i^i  ( x  X.  x ) ) )
119, 10fvmptg 5953 . 2  |-  ( ( A  e.  _V  /\  (  _I  i^i  ( A  X.  A ) )  e.  _V )  -> 
(Diag `  A )  =  (  _I  i^i  ( A  X.  A
) ) )
121, 6, 11syl2anc 665 1  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078    i^i cin 3432    _I cid 4755    X. cxp 4843   ` cfv 5592  Diagcdiag2 31429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-bj-diag 31430
This theorem is referenced by:  bj-eldiag  31432  bj-eldiag2  31433
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