Theorem addsubass 8181

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 998	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 8170	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3	2	3adant1 1016	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4		simp3 1000	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5	1, 3, 4	addassd 7994	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + ( ( B - C ) + C ) ) )
6		npcan 8180	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
7	6	3adant1 1016	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
8	7	oveq2d 5904	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + C ) ) = ( A + B ) )
9	5, 8	eqtrd 2220	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + B ) )
10	9	oveq1d 5903	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( ( A + B ) - C ) )
11	1, 3	addcld 7991	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B - C ) ) e. CC )
12		pncan 8177	. . 3 |- ( ( ( A + ( B - C ) ) e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
13	11, 4, 12	syl2anc 411	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( A + ( B - C ) ) )
14	10, 13	eqtr3d 2222	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158  (class class class)co 5888   CCcc 7823   + caddc 7828   - cmin 8142

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-setind 4548  ax-resscn 7917  ax-1cn 7918  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-sub 8144

This theorem is referenced by:  addsub  8182  subadd23  8183  addsubeq4  8186  npncan  8192  subsub  8201  subsub3  8203  addsub4  8214  negsub  8219  addsubassi  8262  addsubassd  8302  zeo  9372  frecfzen2  10441  odd2np1  11892