Theorem pnncan 7784

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem pnncan

Step	Hyp	Ref	Expression
1		simp1 944	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp2 945	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
3	1, 2	addcld 7568	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
4		simp3 946	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		subsub 7773	. . 3 |- ( ( ( A + B ) e. CC /\ A e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( ( ( A + B ) - A ) + C ) )
6	3, 1, 4, 5	syl3anc 1175	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( ( ( A + B ) - A ) + C ) )
7		pncan2 7750	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
8	7	3adant3 964	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - A ) = B )
9	8	oveq1d 5681	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) + C ) = ( B + C ) )
10	6, 9	eqtrd 2121	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A - C ) ) = ( B + C ) )

Syntax hints:   -> wi 4   /\ w3a 925   = wceq 1290   e. wcel 1439  (class class class)co 5666   CCcc 7409   + caddc 7414   - cmin 7714

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-resscn 7498  ax-1cn 7499  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-addass 7508  ax-distr 7510  ax-i2m1 7511  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7716

This theorem is referenced by:  ppncan  7785  pnncani  7838  pnncand  7893  halfaddsub  8711  shftval2  10321  sinmul  11096