Theorem subval 8334

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 8333	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
2		riotacl 5969	. . . 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 2239	. . . 4 |- ( y = A -> ( ( z + x ) = y <-> ( z + x ) = A ) )
6	5	riotabidv 5955	. . 3 |- ( y = A -> ( iota_ x e. CC ( z + x ) = y ) = ( iota_ x e. CC ( z + x ) = A ) )
7		oveq1 6007	. . . . 5 |- ( z = B -> ( z + x ) = ( B + x ) )
8	7	eqeq1d 2238	. . . 4 |- ( z = B -> ( ( z + x ) = A <-> ( B + x ) = A ) )
9	8	riotabidv 5955	. . 3 |- ( z = B -> ( iota_ x e. CC ( z + x ) = A ) = ( iota_ x e. CC ( B + x ) = A ) )
10		df-sub 8315	. . 3 |- - = ( y e. CC , z e. CC |-> ( iota_ x e. CC ( z + x ) = y ) )
11	6, 9, 10	ovmpog 6138	. 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 1372	1 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   E!wreu 2510   iota_crio 5952  (class class class)co 6000   CCcc 7993   + caddc 7998   - cmin 8313

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628  ax-resscn 8087  ax-1cn 8088  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-sub 8315

This theorem is referenced by:  subcl  8341  subf  8344  subadd  8345