Theorem addltmul 7229

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 7163	. . . . . . 7 |- 2 e. RR
2		1re 6598	. . . . . . 7 |- 1 e. RR
3		ltsub1 6851	. . . . . . 7 |- ((2 e. RR /\ A e. RR /\ 1 e. RR) -> (2 < A <-> (2 - 1) < (A - 1)))
4	1, 2, 3	mp3an13 1182	. . . . . 6 |- (A e. RR -> (2 < A <-> (2 - 1) < (A - 1)))
5		2cn 7164	. . . . . . . 8 |- 2 e. CC
6		ax1cn 6422	. . . . . . . 8 |- 1 e. CC
7		df-2 7154	. . . . . . . . 9 |- 2 = (1 + 1)
8	7	eqcomi 1888	. . . . . . . 8 |- (1 + 1) = 2
9	5, 6, 6, 8	subaddrii 6529	. . . . . . 7 |- (2 - 1) = 1
10	9	breq1i 3345	. . . . . 6 |- ((2 - 1) < (A - 1) <-> 1 < (A - 1))
11	4, 10	syl6bb 595	. . . . 5 |- (A e. RR -> (2 < A <-> 1 < (A - 1)))
12		ltsub1 6851	. . . . . . 7 |- ((2 e. RR /\ B e. RR /\ 1 e. RR) -> (2 < B <-> (2 - 1) < (B - 1)))
13	1, 2, 12	mp3an13 1182	. . . . . 6 |- (B e. RR -> (2 < B <-> (2 - 1) < (B - 1)))
14	9	breq1i 3345	. . . . . 6 |- ((2 - 1) < (B - 1) <-> 1 < (B - 1))
15	13, 14	syl6bb 595	. . . . 5 |- (B e. RR -> (2 < B <-> 1 < (B - 1)))
16	11, 15	bi2anan9 694	. . . 4 |- ((A e. RR /\ B e. RR) -> ((2 < A /\ 2 < B) <-> (1 < (A - 1) /\ 1 < (B - 1))))
17		mulgt1 7027	. . . . . 6 |- ((((A - 1) e. RR /\ (B - 1) e. RR) /\ (1 < (A - 1) /\ 1 < (B - 1))) -> 1 < ((A - 1) x. (B - 1)))
18	17	ex 402	. . . . 5 |- (((A - 1) e. RR /\ (B - 1) e. RR) -> ((1 < (A - 1) /\ 1 < (B - 1)) -> 1 < ((A - 1) x. (B - 1))))
19		peano2rem 6605	. . . . 5 |- (A e. RR -> (A - 1) e. RR)
20		peano2rem 6605	. . . . 5 |- (B e. RR -> (B - 1) e. RR)
21	18, 19, 20	syl2an 503	. . . 4 |- ((A e. RR /\ B e. RR) -> ((1 < (A - 1) /\ 1 < (B - 1)) -> 1 < ((A - 1) x. (B - 1))))
22	16, 21	sylbid 220	. . 3 |- ((A e. RR /\ B e. RR) -> ((2 < A /\ 2 < B) -> 1 < ((A - 1) x. (B - 1))))
23		mulsub 6644	. . . . . . . 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))))
24	6, 23	mpanl2 771	. . . . . . 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))))
25	6, 24	mpanr2 776	. . . . . 6 |- ((A e. CC /\ B e. CC) -> ((A - 1) x. (B - 1)) = (((A x. B) + (1 x. 1)) - ((A x. 1) + (B x. 1))))
26		recn 6466	. . . . . 6 |- (A e. RR -> A e. CC)
27		recn 6466	. . . . . 6 |- (B e. RR -> B e. CC)
28	25, 26, 27	syl2an 503	. . . . 5 |- ((A e. RR /\ B e. RR) -> ((A - 1) x. (B - 1)) = (((A x. B) + (1 x. 1)) - ((A x. 1) + (B x. 1))))
29	28	breq2d 3350	. . . 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)))))
30		readdcl 6455	. . . . . . 7 |- (((A x. 1) e. RR /\ (B x. 1) e. RR) -> ((A x. 1) + (B x. 1)) e. RR)
31		axmulrcl 6427	. . . . . . . 8 |- ((A e. RR /\ 1 e. RR) -> (A x. 1) e. RR)
32	2, 31	mpan2 760	. . . . . . 7 |- (A e. RR -> (A x. 1) e. RR)
33		axmulrcl 6427	. . . . . . . 8 |- ((B e. RR /\ 1 e. RR) -> (B x. 1) e. RR)
34	2, 33	mpan2 760	. . . . . . 7 |- (B e. RR -> (B x. 1) e. RR)
35	30, 32, 34	syl2an 503	. . . . . 6 |- ((A e. RR /\ B e. RR) -> ((A x. 1) + (B x. 1)) e. RR)
36		readdcl 6455	. . . . . . 7 |- (((A x. B) e. RR /\ (1 x. 1) e. RR) -> ((A x. B) + (1 x. 1)) e. RR)
37		axmulrcl 6427	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
38	2, 2	remulcli 6488	. . . . . . 7 |- (1 x. 1) e. RR
39	36, 37, 38	sylancl 525	. . . . . 6 |- ((A e. RR /\ B e. RR) -> ((A x. B) + (1 x. 1)) e. RR)
40		ltaddsub2 6815	. . . . . . 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 1179	. . . . . 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	syl11anc 524	. . . . 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	6	mulid1i 6485	. . . . . . 7 |- (1 x. 1) = 1
44	43	opreq2i 4893	. . . . . 6 |- ((A x. B) + (1 x. 1)) = ((A x. B) + 1)
45	44	breq2i 3346	. . . . 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))
46	42, 45	syl5rbbr 594	. . . 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)))
47		ltadd1 6806	. . . . . . 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)))
48	2, 47	mp3an3 1180	. . . . . 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)))
49	35, 37, 48	syl11anc 524	. . . . 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)))
50		mulid1 6464	. . . . . . . 8 |- (A e. CC -> (A x. 1) = A)
51	26, 50	syl 12	. . . . . . 7 |- (A e. RR -> (A x. 1) = A)
52		mulid1 6464	. . . . . . . 8 |- (B e. CC -> (B x. 1) = B)
53	27, 52	syl 12	. . . . . . 7 |- (B e. RR -> (B x. 1) = B)
54	51, 53	opreqan12d 4902	. . . . . 6 |- ((A e. RR /\ B e. RR) -> ((A x. 1) + (B x. 1)) = (A + B))
55	54	breq1d 3348	. . . . 5 |- ((A e. RR /\ B e. RR) -> (((A x. 1) + (B x. 1)) < (A x. B) <-> (A + B) < (A x. B)))
56	49, 55	bitr3d 589	. . . 4 |- ((A e. RR /\ B e. RR) -> ((((A x. 1) + (B x. 1)) + 1) < ((A x. B) + 1) <-> (A + B) < (A x. B)))
57	29, 46, 56	3bitrd 603	. . 3 |- ((A e. RR /\ B e. RR) -> (1 < ((A - 1) x. (B - 1)) <-> (A + B) < (A x. B)))
58	22, 57	sylibd 219	. 2 |- ((A e. RR /\ B e. RR) -> ((2 < A /\ 2 < B) -> (A + B) < (A x. B)))
59	58	imp 377	1 |- (((A e. RR /\ B e. RR) /\ (2 < A /\ 2 < B)) -> (A + B) < (A x. B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   < clt 6653  2c2 7145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-2 7154

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