addltmul - Intuitionistic Logic Explorer

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Theorem addltmul 9336

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 9168	. . . . . . 7
2		1re 8133	. . . . . . 7
3		ltsub1 8593	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1362	. . . . . 6
5		2m1e1 9216	. . . . . . 7
6	5	breq1i 4089	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8593	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1362	. . . . . 6
10	5	breq1i 4089	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 608	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8401	. . . . 5
14		peano2rem 8401	. . . . 5
15		mulgt1 8998	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 8120	. . . . . 6
20		recn 8120	. . . . . 6
21		ax-1cn 8080	. . . . . . 7
22		mulsub 8535	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4094	. . . 4 $/\$
27		remulcl 8115	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 8115	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 8113	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 8115	. . . . . . 7 $/\$
34	2, 2	remulcli 8148	. . . . . . 7
35		readdcl 8113	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8572	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1359	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9251	. . . . . . 7
41	40	oveq2i 6005	. . . . . 6
42	41	breq2i 4090	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8564	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1360	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 8094	. . . . . . 7
48		ax-1rid 8094	. . . . . . 7
49	47, 48	oveqan12d 6013	. . . . . 6 $/\$
50	49	breq1d 4092	. . . . 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 4082 (class class class)co 5994

cc 7985

cr 7986

c1 7988

caddc 7990

cmul 7992

clt 8169

cmin 8305

c2 9149

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 4201 ax-pow 4257 ax-pr 4292 ax-un 4521 ax-setind 4626 ax-cnex 8078 ax-resscn 8079 ax-1cn 8080 ax-1re 8081 ax-icn 8082 ax-addcl 8083 ax-addrcl 8084 ax-mulcl 8085 ax-mulrcl 8086 ax-addcom 8087 ax-mulcom 8088 ax-addass 8089 ax-mulass 8090 ax-distr 8091 ax-i2m1 8092 ax-0lt1 8093 ax-1rid 8094 ax-0id 8095 ax-rnegex 8096 ax-precex 8097 ax-cnre 8098 ax-pre-lttrn 8101 ax-pre-ltadd 8103 ax-pre-mulgt0 8104

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 3888 df-br 4083 df-opab 4145 df-id 4381 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-dm 4726 df-iota 5274 df-fun 5316 df-fv 5322 df-riota 5947 df-ov 5997 df-oprab 5998 df-mpo 5999 df-pnf 8171 df-mnf 8172 df-ltxr 8174 df-sub 8307 df-neg 8308 df-2 9157

This theorem is referenced by: (None)

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