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Theorem subval 9043
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 2292 . . 3  |-  ( y  =  A  ->  (
( z  +  x
)  =  y  <->  ( z  +  x )  =  A ) )
21riotabidv 6306 . 2  |-  ( y  =  A  ->  ( iota_ x  e.  CC ( z  +  x )  =  y )  =  ( iota_ x  e.  CC ( z  +  x
)  =  A ) )
3 oveq1 5865 . . . 4  |-  ( z  =  B  ->  (
z  +  x )  =  ( B  +  x ) )
43eqeq1d 2291 . . 3  |-  ( z  =  B  ->  (
( z  +  x
)  =  A  <->  ( B  +  x )  =  A ) )
54riotabidv 6306 . 2  |-  ( z  =  B  ->  ( iota_ x  e.  CC ( z  +  x )  =  A )  =  ( iota_ x  e.  CC ( B  +  x
)  =  A ) )
6 df-sub 9039 . 2  |-  -  =  ( y  e.  CC ,  z  e.  CC  |->  ( iota_ x  e.  CC ( z  +  x
)  =  y ) )
7 riotaex 6308 . 2  |-  ( iota_ x  e.  CC ( B  +  x )  =  A )  e.  _V
82, 5, 6, 7ovmpt2 5983 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 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   iota_crio 6297   CCcc 8735    + caddc 8740    - cmin 9037
This theorem is referenced by:  subcl  9051  subf  9053  subadd  9054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-sub 9039
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