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Theorem subval 9289
Description: Value of subtraction, which is the (unique) element  x such that  B  +  x  =  A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
Assertion
Ref Expression
subval  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC ( B  +  x )  =  A ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem subval
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2444 . . 3  |-  ( y  =  A  ->  (
( z  +  x
)  =  y  <->  ( z  +  x )  =  A ) )
21riotabidv 6543 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  CC ( z  +  x )  =  y )  =  ( iota_ x  e.  CC ( z  +  x
)  =  A ) )
3 oveq1 6080 . . . 4  |-  ( z  =  B  ->  (
z  +  x )  =  ( B  +  x ) )
43eqeq1d 2443 . . 3  |-  ( z  =  B  ->  (
( z  +  x
)  =  A  <->  ( B  +  x )  =  A ) )
54riotabidv 6543 . 2  |-  ( z  =  B  ->  ( iota_ x  e.  CC ( z  +  x )  =  A )  =  ( iota_ x  e.  CC ( B  +  x
)  =  A ) )
6 df-sub 9285 . 2  |-  -  =  ( y  e.  CC ,  z  e.  CC  |->  ( iota_ x  e.  CC ( z  +  x
)  =  y ) )
7 riotaex 6545 . 2  |-  ( iota_ x  e.  CC ( B  +  x )  =  A )  e.  _V
82, 5, 6, 7ovmpt2 6201 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  =  ( iota_ x  e.  CC ( B  +  x )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725  (class class class)co 6073   iota_crio 6534   CCcc 8980    + caddc 8985    - cmin 9283
This theorem is referenced by:  subcl  9297  subf  9299  subadd  9300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-sub 9285
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