Theorem addltmul 8956

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 8790	. . . . . . 7 |- 2 e. RR
2		1re 7765	. . . . . . 7 |- 1 e. RR
3		ltsub1 8220	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
4	1, 2, 3	mp3an13 1306	. . . . . 6 |- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
5		2m1e1 8838	. . . . . . 7 |- ( 2 - 1 ) = 1
6	5	breq1i 3936	. . . . . 6 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
7	4, 6	syl6bb 195	. . . . 5 |- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) )
8		ltsub1 8220	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
9	1, 2, 8	mp3an13 1306	. . . . . 6 |- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
10	5	breq1i 3936	. . . . . 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 595	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
13		peano2rem 8029	. . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
14		peano2rem 8029	. . . . 5 |- ( B e. RR -> ( B - 1 ) e. RR )
15		mulgt1 8621	. . . . . 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 287	. . . 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 7753	. . . . . 6 |- ( A e. RR -> A e. CC )
20		recn 7753	. . . . . 6 |- ( B e. RR -> B e. CC )
21		ax-1cn 7713	. . . . . . 7 |- 1 e. CC
22		mulsub 8163	. . . . . . . 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 431	. . . . . . 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 434	. . . . . 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 287	. . . . 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 3941	. . . 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 8872	. . . . . . 7 |- ( 1 x. 1 ) = 1
28	27	oveq2i 5785	. . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
29	28	breq2i 3937	. . . . 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 7748	. . . . . . . 8 |- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR )
31	2, 30	mpan2 421	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) e. RR )
32		remulcl 7748	. . . . . . . 8 |- ( ( B e. RR /\ 1 e. RR ) -> ( B x. 1 ) e. RR )
33	2, 32	mpan2 421	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) e. RR )
34		readdcl 7746	. . . . . . 7 |- ( ( ( A x. 1 ) e. RR /\ ( B x. 1 ) e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
35	31, 33, 34	syl2an 287	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
36		remulcl 7748	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
37	2, 2	remulcli 7780	. . . . . . 7 |- ( 1 x. 1 ) e. RR
38		readdcl 7746	. . . . . . 7 |- ( ( ( A x. B ) e. RR /\ ( 1 x. 1 ) e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
39	36, 37, 38	sylancl 409	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
40		ltaddsub2 8199	. . . . . . 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 1303	. . . . . 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 408	. . . . 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 8191	. . . . . . 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 1304	. . . . . 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 408	. . . . 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 7727	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
48		ax-1rid 7727	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) = B )
49	47, 48	oveqan12d 5793	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
50	49	breq1d 3939	. . . . 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 1331   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   - cmin 7933   2c2 8771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-lttrn 7734  ax-pre-ltadd 7736  ax-pre-mulgt0 7737

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-sub 7935  df-neg 7936  df-2 8779

This theorem is referenced by: (None)