Theorem muleqadd 7431

Description: Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Assertion

Ref	Expression
muleqadd	|- ((A e. CC /\ B e. CC) -> ((A x. B) = (A + B) <-> ((A - 1) x. (B - 1)) = 1))

Proof of Theorem muleqadd

Step	Hyp	Ref	Expression
1		ax-1cn 6776	. . . . 5 |- 1 e. CC
2		mulsub 7194	. . . . . 6 |- (((A e. CC /\ 1 e. CC) /\ (B e. CC /\ 1 e. CC)) -> ((A - 1) x. (B - 1)) = (((A x. B) + ( 1 x. 1)) - ((A x. 1) + (B x. 1))))
3	1, 2	mpanr2 414	. . . . 5 |- (((A e. CC /\ 1 e. CC) /\ B e. CC) -> ((A - 1) x. (B - 1)) = (((A x. B) + ( 1 x. 1)) - ((A x. 1) + (B x. 1))))
4	1, 3	mpanl2 411	. . . 4 |- ((A e. CC /\ B e. CC) -> ((A - 1) x. (B - 1)) = (((A x. B) + ( 1 x. 1)) - ((A x. 1) + (B x. 1))))
5	1	mulid1i 6827	. . . . . . 7 |- ( 1 x. 1) = 1
6	5	oveq2i 5466	. . . . . 6 |- ((A x. B) + ( 1 x. 1)) = ((A x. B) + 1)
7	6	a1i 9	. . . . 5 |- ((A e. CC /\ B e. CC) -> ((A x. B) + ( 1 x. 1)) = ((A x. B) + 1))
8		mulid1 6822	. . . . . 6 |- (A e. CC -> (A x. 1) = A)
9		mulid1 6822	. . . . . 6 |- (B e. CC -> (B x. 1) = B)
10	8, 9	oveqan12d 5474	. . . . 5 |- ((A e. CC /\ B e. CC) -> ((A x. 1) + (B x. 1)) = (A + B))
11	7, 10	oveq12d 5473	. . . 4 |- ((A e. CC /\ B e. CC) -> (((A x. B) + ( 1 x. 1)) - ((A x. 1) + (B x. 1))) = (((A x. B) + 1) - (A + B)))
12		mulcl 6806	. . . . 5 |- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
13		addcl 6804	. . . . 5 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
14		addsub 7019	. . . . . 6 |- (((A x. B) e. CC /\ 1 e. CC /\ (A + B) e. CC) -> (((A x. B) + 1) - (A + B)) = (((A x. B) - (A + B)) + 1))
15	1, 14	mp3an2 1219	. . . . 5 |- (((A x. B) e. CC /\ (A + B) e. CC) -> (((A x. B) + 1) - (A + B)) = (((A x. B) - (A + B)) + 1))
16	12, 13, 15	syl2anc 391	. . . 4 |- ((A e. CC /\ B e. CC) -> (((A x. B) + 1) - (A + B)) = (((A x. B) - (A + B)) + 1))
17	4, 11, 16	3eqtrd 2073	. . 3 |- ((A e. CC /\ B e. CC) -> ((A - 1) x. (B - 1)) = (((A x. B) - (A + B)) + 1))
18	17	eqeq1d 2045	. 2 |- ((A e. CC /\ B e. CC) -> (((A - 1) x. (B - 1)) = 1 <-> (((A x. B) - (A + B)) + 1) = 1))
19	1	addid2i 6953	. . . 4 |- ( 0 + 1) = 1
20	19	eqeq2i 2047	. . 3 |- ((((A x. B) - (A + B)) + 1) = ( 0 + 1) <-> (((A x. B) - (A + B)) + 1) = 1)
21	12, 13	subcld 7118	. . . 4 |- ((A e. CC /\ B e. CC) -> ((A x. B) - (A + B)) e. CC)
22		0cn 6817	. . . . 5 |- 0 e. CC
23		addcan2 6989	. . . . 5 |- ((((A x. B) - (A + B)) e. CC /\ 0 e. CC /\ 1 e. CC) -> ((((A x. B) - (A + B)) + 1) = ( 0 + 1) <-> ((A x. B) - (A + B)) = 0))
24	22, 1, 23	mp3an23 1223	. . . 4 |- (((A x. B) - (A + B)) e. CC -> ((((A x. B) - (A + B)) + 1) = ( 0 + 1) <-> ((A x. B) - (A + B)) = 0))
25	21, 24	syl 14	. . 3 |- ((A e. CC /\ B e. CC) -> ((((A x. B) - (A + B)) + 1) = ( 0 + 1) <-> ((A x. B) - (A + B)) = 0))
26	20, 25	syl5rbbr 184	. 2 |- ((A e. CC /\ B e. CC) -> (((A x. B) - (A + B)) = 0 <-> (((A x. B) - (A + B)) + 1) = 1))
27	12, 13	subeq0ad 7128	. 2 |- ((A e. CC /\ B e. CC) -> (((A x. B) - (A + B)) = 0 <-> (A x. B) = (A + B)))
28	18, 26, 27	3bitr2rd 206	1 |- ((A e. CC /\ B e. CC) -> ((A x. B) = (A + B) <-> ((A - 1) x. (B - 1)) = 1))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390  (class class class)co 5455   CCcc 6709   0cc0 6711   1c1 6712   + caddc 6714   x. cmul 6716   - cmin 6979

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220  ax-resscn 6775  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-sub 6981  df-neg 6982

This theorem is referenced by:  conjmulap  7487