Theorem muleqadd 8627

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 7906	. . . . 5 |- 1 e. CC
2		mulsub 8360	. . . . . 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 7961	. . . . . . 7 |- ( 1 x. 1 ) = 1
6	5	oveq2i 5888	. . . . . 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 7956	. . . . . 6 |- ( A e. CC -> ( A x. 1 ) = A )
9		mulrid 7956	. . . . . 6 |- ( B e. CC -> ( B x. 1 ) = B )
10	8, 9	oveqan12d 5896	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
11	7, 10	oveq12d 5895	. . . 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 7940	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13		addcl 7938	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
14		addsub 8170	. . . . . 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 1325	. . . . 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 2214	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) - ( A + B ) ) + 1 ) )
18	17	eqeq1d 2186	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( A - 1 ) x. ( B - 1 ) ) = 1 <-> ( ( ( A x. B ) - ( A + B ) ) + 1 ) = 1 ) )
19	12, 13	subcld 8270	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) - ( A + B ) ) e. CC )
20		0cn 7951	. . . . 5 |- 0 e. CC
21		addcan2 8140	. . . . 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 1329	. . . 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 8102	. . . 4 |- ( 0 + 1 ) = 1
25	24	eqeq2i 2188	. . 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 8280	. 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 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   - cmin 8130

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538  ax-resscn 7905  ax-1cn 7906  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-sub 8132  df-neg 8133

This theorem is referenced by:  conjmulap  8688