Theorem ppncan 8421

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 8316	. . . 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 6033	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - ( B - C ) ) = ( ( B + A ) - ( B - C ) ) )
4		addcl 8157	. . . 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 8407	. . 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 8420	. . 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 2272	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 2202  (class class class)co 6018   CCcc 8030   + caddc 8035   - cmin 8350

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 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635  ax-resscn 8124  ax-1cn 8125  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-sub 8352

This theorem is referenced by:  ppncand  8530  halfaddsub  9378  pythagtriplem4  12842  pythagtriplem14  12851  ptolemy  15550