Theorem addsubeq4 8134

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 2172	. . 3 |- ( ( C - A ) = ( B - D ) <-> ( B - D ) = ( C - A ) )
2		subcl 8118	. . . . . 6 |- ( ( C e. CC /\ A e. CC ) -> ( C - A ) e. CC )
3	2	ancoms 266	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( C - A ) e. CC )
4		subadd 8122	. . . . . . 7 |- ( ( B e. CC /\ D e. CC /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
5	4	3expa 1198	. . . . . 6 |- ( ( ( B e. CC /\ D e. CC ) /\ ( C - A ) e. CC ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
6	5	ancoms 266	. . . . 5 |- ( ( ( C - A ) e. CC /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
7	3, 6	sylan 281	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
8	7	an4s 583	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( B - D ) = ( C - A ) <-> ( D + ( C - A ) ) = B ) )
9	1, 8	syl5bb 191	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C - A ) = ( B - D ) <-> ( D + ( C - A ) ) = B ) )
10		addcom 8056	. . . . . . 7 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
11	10	adantl 275	. . . . . 6 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( C + D ) = ( D + C ) )
12	11	oveq1d 5868	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( ( D + C ) - A ) )
13		addsubass 8129	. . . . . . . 8 |- ( ( D e. CC /\ C e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
14	13	3com12 1202	. . . . . . 7 |- ( ( C e. CC /\ D e. CC /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
15	14	3expa 1198	. . . . . 6 |- ( ( ( C e. CC /\ D e. CC ) /\ A e. CC ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
16	15	ancoms 266	. . . . 5 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( D + C ) - A ) = ( D + ( C - A ) ) )
17	12, 16	eqtrd 2203	. . . 4 |- ( ( A e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
18	17	adantlr 474	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( C + D ) - A ) = ( D + ( C - A ) ) )
19	18	eqeq1d 2179	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( D + ( C - A ) ) = B ) )
20		addcl 7899	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
21		subadd 8122	. . . . 5 |- ( ( ( C + D ) e. CC /\ A e. CC /\ B e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
22	21	3expb 1199	. . . 4 |- ( ( ( C + D ) e. CC /\ ( A e. CC /\ B e. CC ) ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
23	22	ancoms 266	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( ( C + D ) - A ) = B <-> ( A + B ) = ( C + D ) ) )
24	20, 23	sylan2 284	. 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 216	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) = ( C + D ) <-> ( C - A ) = ( B - D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   - cmin 8090

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092

This theorem is referenced by:  subcan  8174  addsubeq4d  8281