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Theorem addltmul 9144

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 8978	. . . . . . 7
2		1re 7947	. . . . . . 7
3		ltsub1 8405	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1328	. . . . . 6
5		2m1e1 9026	. . . . . . 7
6	5	breq1i 4007	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8405	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1328	. . . . . 6
10	5	breq1i 4007	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 606	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8214	. . . . 5
14		peano2rem 8214	. . . . 5
15		mulgt1 8809	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 7935	. . . . . 6
20		recn 7935	. . . . . 6
21		ax-1cn 7895	. . . . . . 7
22		mulsub 8348	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4012	. . . 4 $/\$
27		remulcl 7930	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 7930	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 7928	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 7930	. . . . . . 7 $/\$
34	2, 2	remulcli 7962	. . . . . . 7
35		readdcl 7928	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8384	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1325	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9060	. . . . . . 7
41	40	oveq2i 5880	. . . . . 6
42	41	breq2i 4008	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8376	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1326	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 7909	. . . . . . 7
48		ax-1rid 7909	. . . . . . 7
49	47, 48	oveqan12d 5888	. . . . . 6 $/\$
50	49	breq1d 4010	. . . . 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 1353

wcel 2148 class class class wbr 4000 (class class class)co 5869

cc 7800

cr 7801

c1 7803

caddc 7805

cmul 7807

clt 7982

cmin 8118

c2 8959

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-lttrn 7916 ax-pre-ltadd 7918 ax-pre-mulgt0 7919

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-ltxr 7987 df-sub 8120 df-neg 8121 df-2 8967

This theorem is referenced by: (None)

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