Theorem pnpcan 8260

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

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

Proof of Theorem pnpcan

Step	Hyp	Ref	Expression
1		simp1 999	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp2 1000	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
3	1, 2	addcld 8041	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
4		simp3 1001	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		subsub4 8254	. . 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 1249	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) - C ) = ( ( A + B ) - ( A + C ) ) )
7		pncan2 8228	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
8	7	3adant3 1019	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - A ) = B )
9	8	oveq1d 5934	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) - C ) = ( B - C ) )
10	6, 9	eqtr3d 2228	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   + caddc 7877   - cmin 8192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194

This theorem is referenced by:  pnpcan2  8261  pnpcand  8369  cju  8982