Theorem addsubass 8129

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 992	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 8118	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
3	2	3adant1 1010	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4		simp3 994	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5	1, 3, 4	addassd 7942	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + ( ( B - C ) + C ) ) )
6		npcan 8128	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
7	6	3adant1 1010	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B - C ) + C ) = B )
8	7	oveq2d 5869	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( ( B - C ) + C ) ) = ( A + B ) )
9	5, 8	eqtrd 2203	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( B - C ) ) + C ) = ( A + B ) )
10	9	oveq1d 5868	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + ( B - C ) ) + C ) - C ) = ( ( A + B ) - C ) )
11	1, 3	addcld 7939	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B - C ) ) e. CC )
12		pncan 8125	. . 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 2205	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( A + ( B - C ) ) )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   - cmin 8090

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092

This theorem is referenced by:  addsub  8130  subadd23  8131  addsubeq4  8134  npncan  8140  subsub  8149  subsub3  8151  addsub4  8162  negsub  8167  addsubassi  8210  addsubassd  8250  zeo  9317  frecfzen2  10383  odd2np1  11832