Theorem addltmul 9544

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 9375	. . . . . . 7 |- 2 e. RR
2		1re 8325	. . . . . . 7 |- 1 e. RR
3		ltsub1 8786	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
4	1, 2, 3	mp3an13 1369	. . . . . 6 |- ( A e. RR -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
5		2m1e1 9423	. . . . . . 7 |- ( 2 - 1 ) = 1
6	5	breq1i 4137	. . . . . 6 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
7	4, 6	bitrdi 196	. . . . 5 |- ( A e. RR -> ( 2 < A <-> 1 < ( A - 1 ) ) )
8		ltsub1 8786	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
9	1, 2, 8	mp3an13 1369	. . . . . 6 |- ( B e. RR -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
10	5	breq1i 4137	. . . . . 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 614	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 2 < A /\ 2 < B ) <-> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
13		peano2rem 8593	. . . . 5 |- ( A e. RR -> ( A - 1 ) e. RR )
14		peano2rem 8593	. . . . 5 |- ( B e. RR -> ( B - 1 ) e. RR )
15		mulgt1 9194	. . . . . 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 8312	. . . . . 6 |- ( A e. RR -> A e. CC )
20		recn 8312	. . . . . 6 |- ( B e. RR -> B e. CC )
21		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
22		mulsub 8728	. . . . . . . 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 439	. . . . . . 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 442	. . . . . 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 4142	. . . 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 8307	. . . . . . . 8 |- ( ( A e. RR /\ 1 e. RR ) -> ( A x. 1 ) e. RR )
28	2, 27	mpan2 429	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) e. RR )
29		remulcl 8307	. . . . . . . 8 |- ( ( B e. RR /\ 1 e. RR ) -> ( B x. 1 ) e. RR )
30	2, 29	mpan2 429	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) e. RR )
31		readdcl 8305	. . . . . . 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 8307	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
34	2, 2	remulcli 8340	. . . . . . 7 |- ( 1 x. 1 ) e. RR
35		readdcl 8305	. . . . . . 7 |- ( ( ( A x. B ) e. RR /\ ( 1 x. 1 ) e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
36	33, 34, 35	sylancl 417	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + ( 1 x. 1 ) ) e. RR )
37		ltaddsub2 8765	. . . . . . 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 1366	. . . . . 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 415	. . . . 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 9458	. . . . . . 7 |- ( 1 x. 1 ) = 1
41	40	oveq2i 6096	. . . . . 6 |- ( ( A x. B ) + ( 1 x. 1 ) ) = ( ( A x. B ) + 1 )
42	41	breq2i 4138	. . . . 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 8757	. . . . . . 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 1367	. . . . . 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 415	. . . . 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 8286	. . . . . . 7 |- ( A e. RR -> ( A x. 1 ) = A )
48		ax-1rid 8286	. . . . . . 7 |- ( B e. RR -> ( B x. 1 ) = B )
49	47, 48	oveqan12d 6104	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. 1 ) + ( B x. 1 ) ) = ( A + B ) )
50	49	breq1d 4140	. . . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   2c2 9356

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-lttrn 8293  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-sub 8499  df-neg 8500  df-2 9364

This theorem is used by: (None)