Theorem muleqadd 7877

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 7183	. . . . 5 |- 1 e. CC
2		mulsub 7624	. . . . . 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 429	. . . . 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 426	. . . 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 7235	. . . . . . 7 |- ( 1 x. 1 ) = 1
6	5	oveq2i 5574	. . . . . 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 7230	. . . . . 6 |- ( A e. CC -> ( A x. 1 ) = A )
9		mulid1 7230	. . . . . 6 |- ( B e. CC -> ( B x. 1 ) = B )
10	8, 9	oveqan12d 5582	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
11	7, 10	oveq12d 5581	. . . 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 7214	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13		addcl 7212	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
14		addsub 7438	. . . . . 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 1257	. . . . 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 403	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
17	4, 11, 16	3eqtrd 2119	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
18	17	eqeq1d 2091	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - 1 ) x. ( B - 1 ) ) = 1 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
19	1	addid2i 7370	. . . 4 |- ( 0 + 1 ) = 1
20	19	eqeq2i 2093	. . 3 |- ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 )
21	12, 13	subcld 7538	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( A + B ) ) e. CC )
22		0cn 7225	. . . . 5 |- 0 e. CC
23		addcan2 7408	. . . . 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 1261	. . . 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 193	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
27	12, 13	subeq0ad 7548	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( A x. B ) = ( A + B ) ) )
28	18, 26, 27	3bitr2rd 215	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   e. wcel 1434  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096   + caddc 7098   x. cmul 7100   - cmin 7398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-resscn 7182  ax-1cn 7183  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400  df-neg 7401

This theorem is referenced by:  conjmulap  7936