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Theorem addltmul 9492

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 9324	. . . . . . 7
2		1re 8289	. . . . . . 7
3		ltsub1 8749	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1365	. . . . . 6
5		2m1e1 9372	. . . . . . 7
6	5	breq1i 4121	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8749	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1365	. . . . . 6
10	5	breq1i 4121	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 610	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8556	. . . . 5
14		peano2rem 8556	. . . . 5
15		mulgt1 9154	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 8276	. . . . . 6
20		recn 8276	. . . . . 6
21		ax-1cn 8236	. . . . . . 7
22		mulsub 8691	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4126	. . . 4 $/\$
27		remulcl 8271	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 8271	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 8269	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 8271	. . . . . . 7 $/\$
34	2, 2	remulcli 8304	. . . . . . 7
35		readdcl 8269	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8728	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1362	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9407	. . . . . . 7
41	40	oveq2i 6069	. . . . . 6
42	41	breq2i 4122	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8720	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1363	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 8250	. . . . . . 7
48		ax-1rid 8250	. . . . . . 7
49	47, 48	oveqan12d 6077	. . . . . 6 $/\$
50	49	breq1d 4124	. . . . 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 1398

wcel 2205 class class class wbr 4114 (class class class)co 6058

cc 8141

cr 8142

c1 8144

caddc 8146

cmul 8148

clt 8324

cmin 8460

c2 9305

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-lttrn 8257 ax-pre-ltadd 8259 ax-pre-mulgt0 8260

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-ltxr 8329 df-sub 8462 df-neg 8463 df-2 9313

This theorem is referenced by: (None)

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