Theorem npncan3 8024

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 982	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 7985	. . . . 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 1001	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - A ) e. CC )
5		simp2 983	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		addsub 7997	. . 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 1217	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
8		pncan3 7994	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
9	8	3adant2 1001	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
10	9	oveq1d 5797	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( C - B ) )
11	7, 10	eqtr3d 2175	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   + caddc 7647   - cmin 7957

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-sub 7959

This theorem is referenced by:  npncan3d  8133