Theorem addsubass 8108

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 987	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 8097	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3	2	3adant1 1005	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4		simp3 989	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5	1, 3, 4	addassd 7921	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + ( ( B - C ) + C ) ) )
6		npcan 8107	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
7	6	3adant1 1005	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
8	7	oveq2d 5858	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + C ) ) = ( A + B ) )
9	5, 8	eqtrd 2198	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + B ) )
10	9	oveq1d 5857	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( ( A + B ) - C ) )
11	1, 3	addcld 7918	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B - C ) ) e. CC )
12		pncan 8104	. . 3 |- ( ( ( A + ( B - C ) ) e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
13	11, 4, 12	syl2anc 409	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
14	10, 13	eqtr3d 2200	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )

Syntax hints:   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756   - cmin 8069

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071

This theorem is referenced by:  addsub  8109  subadd23  8110  addsubeq4  8113  npncan  8119  subsub  8128  subsub3  8130  addsub4  8141  negsub  8146  addsubassi  8189  addsubassd  8229  zeo  9296  frecfzen2  10362  odd2np1  11810