Theorem muleqadd 8689

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 7967	. . . . 5 |- 1 e. CC
2		mulsub 8422	. . . . . 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 438	. . . . 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 435	. . . 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 8023	. . . . . . 7 |- ( 1 x. 1 ) = 1
6	5	oveq2i 5930	. . . . . 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		mulrid 8018	. . . . . 6 |- ( A e. CC -> ( A x. 1 ) = A )
9		mulrid 8018	. . . . . 6 |- ( B e. CC -> ( B x. 1 ) = B )
10	8, 9	oveqan12d 5938	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
11	7, 10	oveq12d 5937	. . . 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 8001	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13		addcl 7999	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
14		addsub 8232	. . . . . 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 1336	. . . . 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 411	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
17	4, 11, 16	3eqtrd 2230	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
18	17	eqeq1d 2202	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - 1 ) x. ( B - 1 ) ) = 1 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
19	12, 13	subcld 8332	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( A + B ) ) e. CC )
20		0cn 8013	. . . . 5 |- 0 e. CC
21		addcan2 8202	. . . . 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 ) )
22	20, 1, 21	mp3an23 1340	. . . 4 |- ( ( ( A x. B ) - ( A + B ) ) e. CC -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) )
23	19, 22	syl 14	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( A x. B ) - ( A + B ) ) = 0 ) )
24	1	addid2i 8164	. . . 4 |- ( 0 + 1 ) = 1
25	24	eqeq2i 2204	. . 3 |- ( ( ( ( A x. B ) - ( A + B ) ) + 1 ) = ( 0 + 1 ) <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 )
26	23, 25	bitr3di 195	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
27	12, 13	subeq0ad 8342	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. B ) - ( A + B ) ) = 0 <-> ( A x. B ) = ( A + B ) ) )
28	18, 26, 27	3bitr2rd 217	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) = ( A + B ) <-> ( ( A - 1 ) x. ( B - 1 ) ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   - cmin 8192

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194  df-neg 8195

This theorem is referenced by:  conjmulap  8750