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Theorem subval 7674
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 7673 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  E! x  e.  CC  ( B  +  x
)  =  A )
2 riotacl 5622 . . . 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 264 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( iota_ x  e.  CC  ( B  +  x
)  =  A )  e.  CC )
5 eqeq2 2097 . . . 4  |-  ( y  =  A  ->  (
( z  +  x
)  =  y  <->  ( z  +  x )  =  A ) )
65riotabidv 5610 . . 3  |-  ( y  =  A  ->  ( iota_ x  e.  CC  (
z  +  x )  =  y )  =  ( iota_ x  e.  CC  ( z  +  x
)  =  A ) )
7 oveq1 5659 . . . . 5  |-  ( z  =  B  ->  (
z  +  x )  =  ( B  +  x ) )
87eqeq1d 2096 . . . 4  |-  ( z  =  B  ->  (
( z  +  x
)  =  A  <->  ( B  +  x )  =  A ) )
98riotabidv 5610 . . 3  |-  ( z  =  B  ->  ( iota_ x  e.  CC  (
z  +  x )  =  A )  =  ( iota_ x  e.  CC  ( B  +  x
)  =  A ) )
10 df-sub 7655 . . 3  |-  -  =  ( y  e.  CC ,  z  e.  CC  |->  ( iota_ x  e.  CC  ( z  +  x
)  =  y ) )
116, 9, 10ovmpt2g 5779 . 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 1274 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 102    = wceq 1289    e. wcel 1438   E!wreu 2361   iota_crio 5607  (class class class)co 5652   CCcc 7348    + caddc 7353    - cmin 7653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655
This theorem is referenced by:  subcl  7681  subf  7684  subadd  7685
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