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Theorem npncan3 8157
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
StepHypRef Expression
1 simp1 992 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 subcl 8118 . . . . 5  |-  ( ( C  e.  CC  /\  A  e.  CC )  ->  ( C  -  A
)  e.  CC )
32ancoms 266 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( C  -  A
)  e.  CC )
433adant2 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 8130 . . 3  |-  ( ( A  e.  CC  /\  ( C  -  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( C  -  A
) )  -  B
)  =  ( ( A  -  B )  +  ( C  -  A ) ) )
71, 4, 5, 6syl3anc 1233 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( C  -  A ) )  -  B )  =  ( ( A  -  B )  +  ( C  -  A
) ) )
8 pncan3 8127 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( C  -  A ) )  =  C )
983adant2 1011 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( C  -  A ) )  =  C )
109oveq1d 5868 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  ( C  -  A ) )  -  B )  =  ( C  -  B ) )
117, 10eqtr3d 2205 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  +  ( C  -  A ) )  =  ( C  -  B ) )
Colors of variables: wff set class
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:  npncan3d  8266
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