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.

Step	Hyp	Ref	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 )
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 2199	. . . 4 |- ( y = A -> ( ( z + x ) = y <-> ( z + x ) = A ) )
6	5	riotabidv 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 ) )
8	7	eqeq1d 2198	. . . 4 |- ( z = B -> ( ( z + x ) = A <-> ( B + x ) = A ) )
9	8	riotabidv 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 ) )
11	6, 9, 10	ovmpog 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 ) )
12	4, 11	mpd3an3 1349	1 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )

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