Theorem addltmul 9424

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 9256	. . . . . . 7 |- 2 e. RR
2		1re 8221	. . . . . . 7 |- 1 e. RR
3		ltsub1 8681	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
4	1, 2, 3	mp3an13 1365	. . . . . 6 |- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
5		2m1e1 9304	. . . . . . 7 |- ( 2 - 1 ) = 1
6	5	breq1i 4100	. . . . . 6 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
7	4, 6	bitrdi 196	. . . . 5 |- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) )
8		ltsub1 8681	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
9	1, 2, 8	mp3an13 1365	. . . . . 6 |- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
10	5	breq1i 4100	. . . . . 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 610	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
13		peano2rem 8489	. . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
14		peano2rem 8489	. . . . 5 |- ( B e. RR -> ( B - 1 ) e. RR )
15		mulgt1 9086	. . . . . 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 8208	. . . . . 6 |- ( A e. RR -> A e. CC )
20		recn 8208	. . . . . 6 |- ( B e. RR -> B e. CC )
21		ax-1cn 8168	. . . . . . 7 |- 1 e. CC
22		mulsub 8623	. . . . . . . 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 4105	. . . 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 8203	. . . . . . . 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 8203	. . . . . . . 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 8201	. . . . . . 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 8203	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
34	2, 2	remulcli 8236	. . . . . . 7 |- ( 1 x. 1 ) e. RR
35		readdcl 8201	. . . . . . 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 8660	. . . . . . 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 1362	. . . . . 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 9339	. . . . . . 7 |- ( 1 x. 1 ) = 1
41	40	oveq2i 6039	. . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
42	41	breq2i 4101	. . . . 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 8652	. . . . . . 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 1363	. . . . . 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 8182	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
48		ax-1rid 8182	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) = B )
49	47, 48	oveqan12d 6047	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
50	49	breq1d 4103	. . . . 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 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8073   RRcr 8074   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8257   - cmin 8393   2c2 9237

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-lttrn 8189  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8259  df-mnf 8260  df-ltxr 8262  df-sub 8395  df-neg 8396  df-2 9245

This theorem is referenced by: (None)