Theorem subval 8246

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.

Step	Hyp	Ref	Expression
1		negeu 8245	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
2		riotacl 5904	. . . 4 |- ( E! x e. CC ( B + x ) = A -> ( iota_ x e. CC ( B + x ) = A ) e. CC )
3	1, 2	syl 14	. . 3 |- ( ( B e. CC /\ A e. CC ) -> ( iota_ x e. CC ( B + x ) = A ) e. CC )
4	3	ancoms 268	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( iota_ x e. CC ( B + x ) = A ) e. CC )
5		eqeq2 2214	. . . 4 |- ( y = A -> ( ( z + x ) = y <-> ( z + x ) = A ) )
6	5	riotabidv 5891	. . 3 |- ( y = A -> ( iota_ x e. CC ( z + x ) = y ) = ( iota_ x e. CC ( z + x ) = A ) )
7		oveq1 5941	. . . . 5 |- ( z = B -> ( z + x ) = ( B + x ) )
8	7	eqeq1d 2213	. . . 4 |- ( z = B -> ( ( z + x ) = A <-> ( B + x ) = A ) )
9	8	riotabidv 5891	. . 3 |- ( z = B -> ( iota_ x e. CC ( z + x ) = A ) = ( iota_ x e. CC ( B + x ) = A ) )
10		df-sub 8227	. . 3 |- - = ( y e. CC , z e. CC |-> ( iota_ x e. CC ( z + x ) = y ) )
11	6, 9, 10	ovmpog 6070	. 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 ) )
12	4, 11	mpd3an3 1350	1 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   E!wreu 2485   iota_crio 5888  (class class class)co 5934   CCcc 7905   + caddc 7910   - cmin 8225

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4583  ax-resscn 7999  ax-1cn 8000  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-addcom 8007  ax-addass 8009  ax-distr 8011  ax-i2m1 8012  ax-0id 8015  ax-rnegex 8016  ax-cnre 8018

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-iota 5229  df-fun 5270  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-sub 8227

This theorem is referenced by:  subcl  8253  subf  8256  subadd  8257