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Theorem addltmul 9245

Description: Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)

**Assertion**
Ref	Expression
addltmul	$/\$ $/\$ $/\$

**Proof of Theorem addltmul**
Step	Hyp	Ref	Expression
1		2re 9077	. . . . . . 7
2		1re 8042	. . . . . . 7
3		ltsub1 8502	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1339	. . . . . 6
5		2m1e1 9125	. . . . . . 7
6	5	breq1i 4041	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8502	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1339	. . . . . 6
10	5	breq1i 4041	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 606	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8310	. . . . 5
14		peano2rem 8310	. . . . 5
15		mulgt1 8907	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 8029	. . . . . 6
20		recn 8029	. . . . . 6
21		ax-1cn 7989	. . . . . . 7
22		mulsub 8444	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4046	. . . 4 $/\$
27		remulcl 8024	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 8024	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 8022	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 8024	. . . . . . 7 $/\$
34	2, 2	remulcli 8057	. . . . . . 7
35		readdcl 8022	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8481	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1336	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9160	. . . . . . 7
41	40	oveq2i 5936	. . . . . 6
42	41	breq2i 4042	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8473	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1337	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 8003	. . . . . . 7
48		ax-1rid 8003	. . . . . . 7
49	47, 48	oveqan12d 5944	. . . . . 6 $/\$
50	49	breq1d 4044	. . . . 5 $/\$
51	46, 50	bitr3d 190	. . . 4 $/\$
52	26, 43, 51	3bitrd 214	. . 3 $/\$
53	18, 52	sylibd 149	. 2 $/\$ $/\$
54	53	imp 124	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2167 class class class wbr 4034 (class class class)co 5925

cc 7894

cr 7895

c1 7897

caddc 7899

cmul 7901

clt 8078

cmin 8214

c2 9058

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-lttrn 8010 ax-pre-ltadd 8012 ax-pre-mulgt0 8013

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-ltxr 8083 df-sub 8216 df-neg 8217 df-2 9066

This theorem is referenced by: (None)

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