Theorem ppncan 8136

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 8031	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2	1	3adant3 1007	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) = ( B + A ) )
3	2	oveq1d 5856	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( B + A ) - ( B - C ) ) )
4		addcl 7874	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
5	4	3adant3 1007	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
6		subsub2 8122	. . 3 |- ( ( ( A + B ) e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
7	5, 6	syld3an1 1274	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
8		pnncan 8135	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
9	8	3com12 1197	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
10	3, 7, 9	3eqtr3d 2206	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + ( C - B ) ) = ( A + C ) )

Syntax hints:   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5841   CCcc 7747   + caddc 7752   - cmin 8065

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-setind 4513  ax-resscn 7841  ax-1cn 7842  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-addcom 7849  ax-addass 7851  ax-distr 7853  ax-i2m1 7854  ax-0id 7857  ax-rnegex 7858  ax-cnre 7860

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-opab 4043  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-iota 5152  df-fun 5189  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-sub 8067

This theorem is referenced by:  ppncand  8245  halfaddsub  9087  pythagtriplem4  12196  pythagtriplem14  12205  ptolemy  13345