Theorem pnpcan 8514

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 1024	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp2 1025	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
3	1, 2	addcld 8295	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
4		simp3 1026	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
5		subsub4 8508	. . 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 1274	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) - C ) = ( ( A + B ) - ( A + C ) ) )
7		pncan2 8482	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - A ) = B )
8	7	3adant3 1044	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - A ) = B )
9	8	oveq1d 6067	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) - A ) - C ) = ( B - C ) )
10	6, 9	eqtr3d 2269	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( A + C ) ) = ( B - C ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6052   CCcc 8127   + caddc 8132   - cmin 8446

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-setind 4661  ax-resscn 8221  ax-1cn 8222  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-sub 8448

This theorem is referenced by:  pnpcan2  8515  pnpcand  8623  cju  9237  swrdswrd  11401  1sgm2ppw  15880