Theorem subval 8284

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 8283	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
2		riotacl 5927	. . . 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 2216	. . . 4 |- ( y = A -> ( ( z + x ) = y <-> ( z + x ) = A ) )
6	5	riotabidv 5914	. . 3 |- ( y = A -> ( iota_ x e. CC ( z + x ) = y ) = ( iota_ x e. CC ( z + x ) = A ) )
7		oveq1 5964	. . . . 5 |- ( z = B -> ( z + x ) = ( B + x ) )
8	7	eqeq1d 2215	. . . 4 |- ( z = B -> ( ( z + x ) = A <-> ( B + x ) = A ) )
9	8	riotabidv 5914	. . 3 |- ( z = B -> ( iota_ x e. CC ( z + x ) = A ) = ( iota_ x e. CC ( B + x ) = A ) )
10		df-sub 8265	. . 3 |- - = ( y e. CC , z e. CC |-> ( iota_ x e. CC ( z + x ) = y ) )
11	6, 9, 10	ovmpog 6093	. 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 1351	1 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   E!wreu 2487   iota_crio 5911  (class class class)co 5957   CCcc 7943   + caddc 7948   - cmin 8263

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593  ax-resscn 8037  ax-1cn 8038  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-sub 8265

This theorem is referenced by:  subcl  8291  subf  8294  subadd  8295