Theorem addsubass 7972

Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem addsubass

Step	Hyp	Ref	Expression
1		simp1 981	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 7961	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3	2	3adant1 999	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4		simp3 983	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5	1, 3, 4	addassd 7788	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + ( ( B - C ) + C ) ) )
6		npcan 7971	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
7	6	3adant1 999	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
8	7	oveq2d 5790	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + C ) ) = ( A + B ) )
9	5, 8	eqtrd 2172	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + B ) )
10	9	oveq1d 5789	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( ( A + B ) - C ) )
11	1, 3	addcld 7785	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B - C ) ) e. CC )
12		pncan 7968	. . 3 |- ( ( ( A + ( B - C ) ) e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
13	11, 4, 12	syl2anc 408	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
14	10, 13	eqtr3d 2174	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )

Syntax hints:   -> wi 4   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   + caddc 7623   - cmin 7933

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935

This theorem is referenced by:  addsub  7973  subadd23  7974  addsubeq4  7977  npncan  7983  subsub  7992  subsub3  7994  addsub4  8005  negsub  8010  addsubassi  8053  addsubassd  8093  zeo  9156  frecfzen2  10200  odd2np1  11570