Theorem npncan3 8226

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 999	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 8187	. . . . 5 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 268	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4	3	3adant2 1018	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - A ) e. CC )
5		simp2 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		addsub 8199	. . 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 1249	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
8		pncan3 8196	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
9	8	3adant2 1018	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
10	9	oveq1d 5912	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( C - B ) )
11	7, 10	eqtr3d 2224	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5897   CCcc 7840   + caddc 7845   - cmin 8159

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7934  ax-1cn 7935  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-sub 8161

This theorem is referenced by:  npncan3d  8335