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Theorem addltmul 9356

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 9188	. . . . . . 7
2		1re 8153	. . . . . . 7
3		ltsub1 8613	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1362	. . . . . 6
5		2m1e1 9236	. . . . . . 7
6	5	breq1i 4090	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8613	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1362	. . . . . 6
10	5	breq1i 4090	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 608	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8421	. . . . 5
14		peano2rem 8421	. . . . 5
15		mulgt1 9018	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 8140	. . . . . 6
20		recn 8140	. . . . . 6
21		ax-1cn 8100	. . . . . . 7
22		mulsub 8555	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4095	. . . 4 $/\$
27		remulcl 8135	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 8135	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 8133	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 8135	. . . . . . 7 $/\$
34	2, 2	remulcli 8168	. . . . . . 7
35		readdcl 8133	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8592	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1359	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9271	. . . . . . 7
41	40	oveq2i 6018	. . . . . 6
42	41	breq2i 4091	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8584	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1360	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 8114	. . . . . . 7
48		ax-1rid 8114	. . . . . . 7
49	47, 48	oveqan12d 6026	. . . . . 6 $/\$
50	49	breq1d 4093	. . . . 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 1395

wcel 2200 class class class wbr 4083 (class class class)co 6007

cc 8005

cr 8006

c1 8008

caddc 8010

cmul 8012

clt 8189

cmin 8325

c2 9169

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-lttrn 8121 ax-pre-ltadd 8123 ax-pre-mulgt0 8124

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8191 df-mnf 8192 df-ltxr 8194 df-sub 8327 df-neg 8328 df-2 9177

This theorem is referenced by: (None)

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