Theorem npncan3 8150

Description: Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
npncan3	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Proof of Theorem npncan3

Step	Hyp	Ref	Expression
1		simp1 992	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		subcl 8111	. . . . 5 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 266	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4	3	3adant2 1011	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C - A ) e. CC )
5		simp2 993	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
6		addsub 8123	. . 3 |- ( ( A e. CC /\ ( C - A ) e. CC /\ B e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
7	1, 4, 5, 6	syl3anc 1233	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( ( A - B ) + ( C - A ) ) )
8		pncan3 8120	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
9	8	3adant2 1011	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( C - A ) ) = C )
10	9	oveq1d 5866	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + ( C - A ) ) - B ) = ( C - B ) )
11	7, 10	eqtr3d 2205	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + ( C - A ) ) = ( C - B ) )

Syntax hints:   -> wi 4   /\ w3a 973   = wceq 1348   e. wcel 2141  (class class class)co 5851   CCcc 7765   + caddc 7770   - cmin 8083

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 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-resscn 7859  ax-1cn 7860  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-addcom 7867  ax-addass 7869  ax-distr 7871  ax-i2m1 7872  ax-0id 7875  ax-rnegex 7876  ax-cnre 7878

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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-sub 8085

This theorem is referenced by:  npncan3d  8259