Theorem npncan3 7993

Description: Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
npncan3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Proof of Theorem npncan3

Step	Hyp	Ref	Expression
1		simp1 981	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 7954	. . . . 5 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 266	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4	3	3adant2 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - A ) e. CC )
5		simp2 982	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		addsub 7966	. . 3 |- ( ( A e. CC /\ ( C - A ) e. CC /\ B e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
7	1, 4, 5, 6	syl3anc 1216	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
8		pncan3 7963	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
9	8	3adant2 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
10	9	oveq1d 5782	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( C - B ) )
11	7, 10	eqtr3d 2172	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Syntax hints:   -> wi 4   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5767   CCcc 7611   + caddc 7616   - cmin 7926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-sub 7928

This theorem is referenced by:  npncan3d  8102