Theorem ppncan 8426

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 8321	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
2	1	3adant3 1043	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) = ( B + A ) )
3	2	oveq1d 6038	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( B + A ) - ( B - C ) ) )
4		addcl 8162	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
5	4	3adant3 1043	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + B ) e. CC )
6		subsub2 8412	. . 3 |- ( ( ( A + B ) e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
7	5, 6	syld3an1 1319	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( A + B ) + ( C - B ) ) )
8		pnncan 8425	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
9	8	3com12 1233	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B + A ) - ( B - C ) ) = ( A + C ) )
10	3, 7, 9	3eqtr3d 2271	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + ( C - B ) ) = ( A + C ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2201  (class class class)co 6023   CCcc 8035   + caddc 8040   - cmin 8355

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-setind 4637  ax-resscn 8129  ax-1cn 8130  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-distr 8141  ax-i2m1 8142  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-sub 8357

This theorem is referenced by:  ppncand  8535  halfaddsub  9383  pythagtriplem4  12864  pythagtriplem14  12873  ptolemy  15577