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Theorem subval 8178
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 negeu 8177 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
2 riotacl 5865 . . . 4  |-  ( E! x  e.  CC  ( B  +  x )  =  A  ->  ( iota_ x  e.  CC  ( B  +  x )  =  A )  e.  CC )
31, 2syl 14 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( iota_ x  e.  CC  ( B  +  x
)  =  A )  e.  CC )
43ancoms 268 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( iota_ x  e.  CC  ( B  +  x
)  =  A )  e.  CC )
5 eqeq2 2199 . . . 4  |-  ( y  =  A  ->  (
( z  +  x
)  =  y  <->  ( z  +  x )  =  A ) )
65riotabidv 5853 . . 3  |-  ( y  =  A  ->  ( iota_ x  e.  CC  (
z  +  x )  =  y )  =  ( iota_ x  e.  CC  ( z  +  x
)  =  A ) )
7 oveq1 5902 . . . . 5  |-  ( z  =  B  ->  (
z  +  x )  =  ( B  +  x ) )
87eqeq1d 2198 . . . 4  |-  ( z  =  B  ->  (
( z  +  x
)  =  A  <->  ( B  +  x )  =  A ) )
98riotabidv 5853 . . 3  |-  ( z  =  B  ->  ( iota_ x  e.  CC  (
z  +  x )  =  A )  =  ( iota_ x  e.  CC  ( B  +  x
)  =  A ) )
10 df-sub 8159 . . 3  |-  -  =  ( y  e.  CC ,  z  e.  CC  |->  ( iota_ x  e.  CC  ( z  +  x
)  =  y ) )
116, 9, 10ovmpog 6030 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( iota_ x  e.  CC  ( B  +  x )  =  A )  e.  CC )  ->  ( A  -  B )  =  (
iota_ x  e.  CC  ( B  +  x
)  =  A ) )
124, 11mpd3an3 1349 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 104    = wceq 1364    e. wcel 2160   E!wreu 2470   iota_crio 5850  (class class class)co 5895   CCcc 7838    + caddc 7843    - cmin 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7932  ax-1cn 7933  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-distr 7944  ax-i2m1 7945  ax-0id 7948  ax-rnegex 7949  ax-cnre 7951
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-sub 8159
This theorem is referenced by:  subcl  8185  subf  8188  subadd  8189
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