Theorem addltmul 9151

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 8985	. . . . . . 7 |- 2 e. RR
2		1re 7953	. . . . . . 7 |- 1 e. RR
3		ltsub1 8411	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
4	1, 2, 3	mp3an13 1328	. . . . . 6 |- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
5		2m1e1 9033	. . . . . . 7 |- ( 2 - 1 ) = 1
6	5	breq1i 4009	. . . . . 6 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
7	4, 6	bitrdi 196	. . . . 5 |- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) )
8		ltsub1 8411	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
9	1, 2, 8	mp3an13 1328	. . . . . 6 |- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
10	5	breq1i 4009	. . . . . 6 |- ( ( 2 - 1 ) < ( B - 1 ) <-> 1 < ( B - 1 ) )
11	9, 10	bitrdi 196	. . . . 5 |- ( B e. RR -> ( 2 < B <-> 1 < ( B - 1 ) ) )
12	7, 11	bi2anan9 606	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
13		peano2rem 8220	. . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
14		peano2rem 8220	. . . . 5 |- ( B e. RR -> ( B - 1 ) e. RR )
15		mulgt1 8816	. . . . . 6 |- ( ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) )
16	15	ex 115	. . . . 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 289	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
18	12, 17	sylbid 150	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
19		recn 7941	. . . . . 6 |- ( A e. RR -> A e. CC )
20		recn 7941	. . . . . 6 |- ( B e. RR -> B e. CC )
21		ax-1cn 7901	. . . . . . 7 |- 1 e. CC
22		mulsub 8354	. . . . . . . 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 435	. . . . . . 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 438	. . . . . 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 289	. . . . 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 4014	. . . 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		remulcl 7936	. . . . . . . 8 |- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR )
28	2, 27	mpan2 425	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) e. RR )
29		remulcl 7936	. . . . . . . 8 |- ( ( B e. RR /\ 1 e. RR ) -> ( B x. 1 ) e. RR )
30	2, 29	mpan2 425	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) e. RR )
31		readdcl 7934	. . . . . . 7 |- ( ( ( A x. 1 ) e. RR /\ ( B x. 1 ) e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
32	28, 30, 31	syl2an 289	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) e. RR )
33		remulcl 7936	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
34	2, 2	remulcli 7968	. . . . . . 7 |- ( 1 x. 1 ) e. RR
35		readdcl 7934	. . . . . . 7 |- ( ( ( A x. B ) e. RR /\ ( 1 x. 1 ) e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
36	33, 34, 35	sylancl 413	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
37		ltaddsub2 8390	. . . . . . 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 ) ) ) ) )
38	2, 37	mp3an2 1325	. . . . . 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 ) ) ) ) )
39	32, 36, 38	syl2anc 411	. . . . 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 ) ) ) ) )
40		1t1e1 9067	. . . . . . 7 |- ( 1 x. 1 ) = 1
41	40	oveq2i 5883	. . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
42	41	breq2i 4010	. . . . 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 ) )
43	39, 42	bitr3di 195	. . . 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 8382	. . . . . . 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 1326	. . . . . 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	32, 33, 45	syl2anc 411	. . . . 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 7915	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
48		ax-1rid 7915	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) = B )
49	47, 48	oveqan12d 5891	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
50	49	breq1d 4012	. . . . 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 190	. . . 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 214	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> ( A + B ) < ( A x. B ) ) )
53	18, 52	sylibd 149	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) -> ( A + B ) < ( A x. B ) ) )
54	53	imp 124	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   class class class wbr 4002  (class class class)co 5872   CCcc 7806   RRcr 7807   1c1 7809   + caddc 7811   x. cmul 7813   < clt 7988   - cmin 8124   2c2 8966

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-mulrcl 7907  ax-addcom 7908  ax-mulcom 7909  ax-addass 7910  ax-mulass 7911  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-1rid 7915  ax-0id 7916  ax-rnegex 7917  ax-precex 7918  ax-cnre 7919  ax-pre-lttrn 7922  ax-pre-ltadd 7924  ax-pre-mulgt0 7925

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-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-sub 8126  df-neg 8127  df-2 8974

This theorem is referenced by: (None)