Theorem addsubeq4 8541

Description: Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Assertion

Ref	Expression
addsubeq4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )

Proof of Theorem addsubeq4

Step	Hyp	Ref	Expression
1		eqcom 2240	. . 3 |- ( ( C - A ) = ( B - D ) <-> ( B - D ) = ( C - A ) )
2		subcl 8525	. . . . . 6 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 268	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4		subadd 8529	. . . . . . 7 |- ( ( B e. CC /\ D e. CC /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
5	4	3expa 1234	. . . . . 6 |- ( ( ( B e. CC /\ D e. CC ) /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
6	5	ancoms 268	. . . . 5 |- ( ( ( C - A ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
7	3, 6	sylan 283	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
8	7	an4s 596	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
9	1, 8	bitrid 192	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C - A ) = ( B - D ) <-> ( D + ( C - A ) ) = B ) )
10		addcom 8463	. . . . . . 7 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
11	10	adantl 277	. . . . . 6 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( C + D ) = ( D + C ) )
12	11	oveq1d 6100	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( ( D + C ) - A ) )
13		addsubass 8536	. . . . . . . 8 |- ( ( D e. CC /\ C e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
14	13	3com12 1238	. . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
15	14	3expa 1234	. . . . . 6 |- ( ( ( C e. CC /\ D e. CC ) /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
16	15	ancoms 268	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
17	12, 16	eqtrd 2271	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
18	17	adantlr 481	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
19	18	eqeq1d 2247	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( D + ( C - A ) ) = B ) )
20		addcl 8304	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
21		subadd 8529	. . . . 5 |- ( ( ( C + D ) e. CC /\ A e. CC /\ B e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
22	21	3expb 1235	. . . 4 |- ( ( ( C + D ) e. CC /\ ( A e. CC /\ B e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
23	22	ancoms 268	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
24	20, 23	sylan2 286	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
25	9, 19, 24	3bitr2rd 217	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   + caddc 8182   - cmin 8497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-sub 8499

This theorem is used by:  subcan  8581  addsubeq4d  8688