Theorem addltmul 8854

Description: Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)

Assertion

Ref	Expression
addltmul	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )

Proof of Theorem addltmul

Step	Hyp	Ref	Expression
1		2re 8694	. . . . . . 7 |- 2 e. RR
2		1re 7683	. . . . . . 7 |- 1 e. RR
3		ltsub1 8133	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
4	1, 2, 3	mp3an13 1287	. . . . . 6 |- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
5		2m1e1 8742	. . . . . . 7 |- ( 2 - 1 ) = 1
6	5	breq1i 3900	. . . . . 6 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
7	4, 6	syl6bb 195	. . . . 5 |- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) )
8		ltsub1 8133	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
9	1, 2, 8	mp3an13 1287	. . . . . 6 |- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
10	5	breq1i 3900	. . . . . 6 |- ( ( 2 - 1 ) < ( B - 1 ) <-> 1 < ( B - 1 ) )
11	9, 10	syl6bb 195	. . . . 5 |- ( B e. RR -> ( 2 < B <-> 1 < ( B - 1 ) ) )
12	7, 11	bi2anan9 578	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
13		peano2rem 7946	. . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
14		peano2rem 7946	. . . . 5 |- ( B e. RR -> ( B - 1 ) e. RR )
15		mulgt1 8525	. . . . . 6 |- ( ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) )
16	15	ex 114	. . . . 5 |- ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
17	13, 14, 16	syl2an 285	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
18	12, 17	sylbid 149	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
19		recn 7671	. . . . . 6 |- ( A e. RR -> A e. CC )
20		recn 7671	. . . . . 6 |- ( B e. RR -> B e. CC )
21		ax-1cn 7632	. . . . . . 7 |- 1 e. CC
22		mulsub 8076	. . . . . . . 8 |- ( ( ( 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 ) ) ) )
23	21, 22	mpanl2 429	. . . . . . 7 |- ( ( A 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 ) ) ) )
24	21, 23	mpanr2 432	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
25	19, 20, 24	syl2an 285	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
26	25	breq2d 3905	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
27		1t1e1 8770	. . . . . . 7 |- ( 1 x. 1 ) = 1
28	27	oveq2i 5737	. . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
29	28	breq2i 3901	. . . . 5 |- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) )
30		remulcl 7666	. . . . . . . 8 |- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR )
31	2, 30	mpan2 419	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) e. RR )
32		remulcl 7666	. . . . . . . 8 |- ( ( B e. RR /\ 1 e. RR ) -> ( B x. 1 ) e. RR )
33	2, 32	mpan2 419	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) e. RR )
34		readdcl 7664	. . . . . . 7 |- ( ( ( A x. 1 ) e. RR /\ ( B x. 1 ) e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
35	31, 33, 34	syl2an 285	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
36		remulcl 7666	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
37	2, 2	remulcli 7698	. . . . . . 7 |- ( 1 x. 1 ) e. RR
38		readdcl 7664	. . . . . . 7 |- ( ( ( A x. B ) e. RR /\ ( 1 x. 1 ) e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
39	36, 37, 38	sylancl 407	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
40		ltaddsub2 8112	. . . . . . 7 |- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ 1 e. RR /\ ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
41	2, 40	mp3an2 1284	. . . . . 6 |- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( ( A x. B ) + ( 1 x. 1 ) ) e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
42	35, 39, 41	syl2anc 406	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + ( 1 x. 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
43	29, 42	syl5rbbr 194	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) )
44		ltadd1 8104	. . . . . . 7 |- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( A x. B ) e. RR /\ 1 e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) )
45	2, 44	mp3an3 1285	. . . . . 6 |- ( ( ( ( A x. 1 ) + ( B x. 1 ) ) e. RR /\ ( A x. B ) e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) )
46	35, 36, 45	syl2anc 406	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) ) )
47		ax-1rid 7646	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
48		ax-1rid 7646	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) = B )
49	47, 48	oveqan12d 5745	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
50	49	breq1d 3903	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A x. 1 ) + ( B x. 1 ) ) < ( A x. B ) <-> ( A + B ) < ( A x. B ) ) )
51	46, 50	bitr3d 189	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( A x. 1 ) + ( B x. 1 ) ) + 1 ) < ( ( A x. B ) + 1 ) <-> ( A + B ) < ( A x. B ) ) )
52	26, 43, 51	3bitrd 213	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> ( A + B ) < ( A x. B ) ) )
53	18, 52	sylibd 148	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> ( A + B ) < ( A x. B ) ) )
54	53	imp 123	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   RRcr 7540   1c1 7542   + caddc 7544   x. cmul 7546   < clt 7718   - cmin 7850   2c2 8675

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-lttrn 7653  ax-pre-ltadd 7655  ax-pre-mulgt0 7656

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-ltxr 7723  df-sub 7852  df-neg 7853  df-2 8683

This theorem is referenced by: (None)