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Theorem cnambpcma 32153
Description: ((a-b)+c)-a = c-a holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
Assertion
Ref Expression
cnambpcma  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  C
)  -  A )  =  ( C  -  B ) )

Proof of Theorem cnambpcma
StepHypRef Expression
1 subcl 9824 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
213adant3 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  B )  e.  CC )
3 simp3 999 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
4 simp1 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
52, 3, 4addsubd 9957 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  C
)  -  A )  =  ( ( ( A  -  B )  -  A )  +  C ) )
6 simpl 457 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
7 simpr 461 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  B  e.  CC )
86, 7, 63jca 1177 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )
)
983adant3 1017 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC ) )
10 sub32 9858 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  A  e.  CC )  ->  (
( A  -  B
)  -  A )  =  ( ( A  -  A )  -  B ) )
119, 10syl 16 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  -  A )  =  ( ( A  -  A )  -  B ) )
1211oveq1d 6296 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  -  A
)  +  C )  =  ( ( ( A  -  A )  -  B )  +  C ) )
13 subcl 9824 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  -  A
)  e.  CC )
1413anidms 645 . . . . 5  |-  ( A  e.  CC  ->  ( A  -  A )  e.  CC )
15143ad2ant1 1018 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  A )  e.  CC )
16 simp2 998 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  B  e.  CC )
1715, 16, 3subadd23d 9958 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  A )  -  B
)  +  C )  =  ( ( A  -  A )  +  ( C  -  B
) ) )
18 subid 9843 . . . . 5  |-  ( A  e.  CC  ->  ( A  -  A )  =  0 )
1918oveq1d 6296 . . . 4  |-  ( A  e.  CC  ->  (
( A  -  A
)  +  ( C  -  B ) )  =  ( 0  +  ( C  -  B
) ) )
20193ad2ant1 1018 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  A
)  +  ( C  -  B ) )  =  ( 0  +  ( C  -  B
) ) )
21 subcl 9824 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  -  B
)  e.  CC )
2221ancoms 453 . . . . 5  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  -  B
)  e.  CC )
2322addid2d 9784 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( 0  +  ( C  -  B ) )  =  ( C  -  B ) )
24233adant1 1015 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
0  +  ( C  -  B ) )  =  ( C  -  B ) )
2517, 20, 243eqtrd 2488 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  A )  -  B
)  +  C )  =  ( C  -  B ) )
265, 12, 253eqtrd 2488 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  -  B )  +  C
)  -  A )  =  ( C  -  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804  (class class class)co 6281   CCcc 9493   0cc0 9495    + caddc 9498    - cmin 9810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636  df-sub 9812
This theorem is referenced by: (None)
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