Theorem subval 10272

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		eqeq2 2633	. . 3 |- ( y = A -> ( ( z + x ) = y <-> ( z + x ) = A ) )
2	1	riotabidv 6613	. 2 |- ( y = A -> ( iota_ x e. CC ( z + x ) = y ) = ( iota_ x e. CC ( z + x ) = A ) )
3		oveq1 6657	. . . 4 |- ( z = B -> ( z + x ) = ( B + x ) )
4	3	eqeq1d 2624	. . 3 |- ( z = B -> ( ( z + x ) = A <-> ( B + x ) = A ) )
5	4	riotabidv 6613	. 2 |- ( z = B -> ( iota_ x e. CC ( z + x ) = A ) = ( iota_ x e. CC ( B + x ) = A ) )
6		df-sub 10268	. 2 |- - = ( y e. CC , z e. CC |-> ( iota_ x e. CC ( z + x ) = y ) )
7		riotaex 6615	. 2 |- ( iota_ x e. CC ( B + x ) = A ) e. _V
8	2, 5, 6, 7	ovmpt2 6796	1 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   iota_crio 6610  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sub 10268

This theorem is referenced by:  subcl  10280  subf  10283  subadd  10284