Theorem subadd 8162

Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Assertion

Ref	Expression
subadd	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( B + C ) = A ) )

Proof of Theorem subadd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subval 8151	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) = ( iota_ x e. CC ( B + x ) = A ) )
2	1	eqeq1d 2186	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
3	2	3adant3 1017	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
4		negeu 8150	. . . . 5 |- ( ( B e. CC /\ A e. CC ) -> E! x e. CC ( B + x ) = A )
5		oveq2 5885	. . . . . . 7 |- ( x = C -> ( B + x ) = ( B + C ) )
6	5	eqeq1d 2186	. . . . . 6 |- ( x = C -> ( ( B + x ) = A <-> ( B + C ) = A ) )
7	6	riota2 5855	. . . . 5 |- ( ( C e. CC /\ E! x e. CC ( B + x ) = A ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
8	4, 7	sylan2 286	. . . 4 |- ( ( C e. CC /\ ( B e. CC /\ A e. CC ) ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
9	8	3impb 1199	. . 3 |- ( ( C e. CC /\ B e. CC /\ A e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
10	9	3com13 1208	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + C ) = A <-> ( iota_ x e. CC ( B + x ) = A ) = C ) )
11	3, 10	bitr4d 191	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) = C <-> ( B + C ) = A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   E!wreu 2457   iota_crio 5832  (class class class)co 5877   CCcc 7811   + caddc 7816   - cmin 8130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132

This theorem is referenced by:  subadd2  8163  subsub23  8164  pncan  8165  pncan3  8167  addsubeq4  8174  subsub2  8187  renegcl  8220  subaddi  8246  subaddd  8288  fzen  10045  nn0ennn  10435  cos2t  11760  cos2tsin  11761  odd2np1  11880  divalgb  11932  sincosq1eq  14345  coskpi  14354