addltmul - Intuitionistic Logic Explorer

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Theorem addltmul 9381

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 9213	. . . . . . 7
2		1re 8178	. . . . . . 7
3		ltsub1 8638	. . . . . . 7 $/\$ $/\$
4	1, 2, 3	mp3an13 1364	. . . . . 6
5		2m1e1 9261	. . . . . . 7
6	5	breq1i 4095	. . . . . 6
7	4, 6	bitrdi 196	. . . . 5
8		ltsub1 8638	. . . . . . 7 $/\$ $/\$
9	1, 2, 8	mp3an13 1364	. . . . . 6
10	5	breq1i 4095	. . . . . 6
11	9, 10	bitrdi 196	. . . . 5
12	7, 11	bi2anan9 610	. . . 4 $/\$ $/\$ $/\$
13		peano2rem 8446	. . . . 5
14		peano2rem 8446	. . . . 5
15		mulgt1 9043	. . . . . 6 $/\$ $/\$ $/\$
16	15	ex 115	. . . . 5 $/\$ $/\$
17	13, 14, 16	syl2an 289	. . . 4 $/\$ $/\$
18	12, 17	sylbid 150	. . 3 $/\$ $/\$
19		recn 8165	. . . . . 6
20		recn 8165	. . . . . 6
21		ax-1cn 8125	. . . . . . 7
22		mulsub 8580	. . . . . . . 8 $/\$ $/\$ $/\$
23	21, 22	mpanl2 435	. . . . . . 7 $/\$ $/\$
24	21, 23	mpanr2 438	. . . . . 6 $/\$
25	19, 20, 24	syl2an 289	. . . . 5 $/\$
26	25	breq2d 4100	. . . 4 $/\$
27		remulcl 8160	. . . . . . . 8 $/\$
28	2, 27	mpan2 425	. . . . . . 7
29		remulcl 8160	. . . . . . . 8 $/\$
30	2, 29	mpan2 425	. . . . . . 7
31		readdcl 8158	. . . . . . 7 $/\$
32	28, 30, 31	syl2an 289	. . . . . 6 $/\$
33		remulcl 8160	. . . . . . 7 $/\$
34	2, 2	remulcli 8193	. . . . . . 7
35		readdcl 8158	. . . . . . 7 $/\$
36	33, 34, 35	sylancl 413	. . . . . 6 $/\$
37		ltaddsub2 8617	. . . . . . 7 $/\$ $/\$
38	2, 37	mp3an2 1361	. . . . . 6 $/\$
39	32, 36, 38	syl2anc 411	. . . . 5 $/\$
40		1t1e1 9296	. . . . . . 7
41	40	oveq2i 6029	. . . . . 6
42	41	breq2i 4096	. . . . 5
43	39, 42	bitr3di 195	. . . 4 $/\$
44		ltadd1 8609	. . . . . . 7 $/\$ $/\$
45	2, 44	mp3an3 1362	. . . . . 6 $/\$
46	32, 33, 45	syl2anc 411	. . . . 5 $/\$
47		ax-1rid 8139	. . . . . . 7
48		ax-1rid 8139	. . . . . . 7
49	47, 48	oveqan12d 6037	. . . . . 6 $/\$
50	49	breq1d 4098	. . . . 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 1397

wcel 2202 class class class wbr 4088 (class class class)co 6018

cc 8030

cr 8031

c1 8033

caddc 8035

cmul 8037

clt 8214

cmin 8350

c2 9194

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-lttrn 8146 ax-pre-ltadd 8148 ax-pre-mulgt0 8149

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-ltxr 8219 df-sub 8352 df-neg 8353 df-2 9202

This theorem is referenced by: (None)

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