Theorem ppncan 8161

Description: Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)

Assertion

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

Proof of Theorem ppncan

Step	Hyp	Ref	Expression
1		addcom 8056	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2	1	3adant3 1012	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) = ( B + A ) )
3	2	oveq1d 5868	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( B + A ) - ( B - C ) ) )
4		addcl 7899	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
5	4	3adant3 1012	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
6		subsub2 8147	. . 3 |- ( ( ( A + B ) e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
7	5, 6	syld3an1 1279	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
8		pnncan 8160	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
9	8	3com12 1202	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
10	3, 7, 9	3eqtr3d 2211	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + ( C - B ) ) = ( A + 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:  ppncand  8270  halfaddsub  9112  pythagtriplem4  12222  pythagtriplem14  12231  ptolemy  13539