ltoddhalfle - Metamath Proof Explorer

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Theorem ltoddhalfle 15085

Description: An integer is less than half of an odd number iff it is less than or equal to the half of the predecessor of the odd number (which is an even number). (Contributed by AV, 29-Jun-2021.)

**Assertion**
Ref	Expression
ltoddhalfle	$/\$ $/\$

Proof of Theorem ltoddhalfle Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odd2np1 15065	. . 3
2		halfre 11246	. . . . . . . . . . . . . . . 16
3	2	a1i 11	. . . . . . . . . . . . . . 15
4		1red 10055	. . . . . . . . . . . . . . 15
5		zre 11381	. . . . . . . . . . . . . . 15
6	3, 4, 5	3jca 1242	. . . . . . . . . . . . . 14 $/\$ $/\$
7	6	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
8		halflt1 11250	. . . . . . . . . . . . 13
9		axltadd 10111	. . . . . . . . . . . . 13 $/\$ $/\$
10	7, 8, 9	mpisyl 21	. . . . . . . . . . . 12 $/\$
11		zre 11381	. . . . . . . . . . . . . 14
12	11	adantl 482	. . . . . . . . . . . . 13 $/\$
13	5, 3	readdcld 10069	. . . . . . . . . . . . . 14
14	13	adantr 481	. . . . . . . . . . . . 13 $/\$
15		peano2z 11418	. . . . . . . . . . . . . . 15
16	15	zred 11482	. . . . . . . . . . . . . 14
17	16	adantr 481	. . . . . . . . . . . . 13 $/\$
18		lttr 10114	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
19	12, 14, 17, 18	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$
20	10, 19	mpan2d 710	. . . . . . . . . . 11 $/\$
21		zleltp1 11428	. . . . . . . . . . . 12 $/\$
22	21	ancoms 469	. . . . . . . . . . 11 $/\$
23	20, 22	sylibrd 249	. . . . . . . . . 10 $/\$
24		halfgt0 11248	. . . . . . . . . . . 12
25	3, 5	jca 554	. . . . . . . . . . . . . 14 $/\$
26	25	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$
27		ltaddpos 10518	. . . . . . . . . . . . 13 $/\$
28	26, 27	syl 17	. . . . . . . . . . . 12 $/\$
29	24, 28	mpbii 223	. . . . . . . . . . 11 $/\$
30	5	adantr 481	. . . . . . . . . . . 12 $/\$
31		lelttr 10128	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
32	12, 30, 14, 31	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$
33	29, 32	mpan2d 710	. . . . . . . . . 10 $/\$
34	23, 33	impbid 202	. . . . . . . . 9 $/\$
35		zcn 11382	. . . . . . . . . . . 12
36		1cnd 10056	. . . . . . . . . . . 12
37		2cnne0 11242	. . . . . . . . . . . . 13 $/\$
38	37	a1i 11	. . . . . . . . . . . 12 $/\$
39		muldivdir 10720	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
40	35, 36, 38, 39	syl3anc 1326	. . . . . . . . . . 11
41	40	breq2d 4665	. . . . . . . . . 10
42	41	adantr 481	. . . . . . . . 9 $/\$
43		2z 11409	. . . . . . . . . . . . . . . . 17
44	43	a1i 11	. . . . . . . . . . . . . . . 16
45		id 22	. . . . . . . . . . . . . . . 16
46	44, 45	zmulcld 11488	. . . . . . . . . . . . . . 15
47	46	zcnd 11483	. . . . . . . . . . . . . 14
48	47	adantr 481	. . . . . . . . . . . . 13 $/\$
49		pncan1 10454	. . . . . . . . . . . . 13
50	48, 49	syl 17	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 6665	. . . . . . . . . . 11 $/\$
52		2cnd 11093	. . . . . . . . . . . . 13
53		2ne0 11113	. . . . . . . . . . . . . 14
54	53	a1i 11	. . . . . . . . . . . . 13
55	35, 52, 54	divcan3d 10806	. . . . . . . . . . . 12
56	55	adantr 481	. . . . . . . . . . 11 $/\$
57	51, 56	eqtrd 2656	. . . . . . . . . 10 $/\$
58	57	breq2d 4665	. . . . . . . . 9 $/\$
59	34, 42, 58	3bitr4d 300	. . . . . . . 8 $/\$
60		oveq1 6657	. . . . . . . . . 10
61	60	breq2d 4665	. . . . . . . . 9
62		oveq1 6657	. . . . . . . . . . 11
63	62	oveq1d 6665	. . . . . . . . . 10
64	63	breq2d 4665	. . . . . . . . 9
65	61, 64	bibi12d 335	. . . . . . . 8
66	59, 65	syl5ibcom 235	. . . . . . 7 $/\$
67	66	ex 450	. . . . . 6
68	67	adantl 482	. . . . 5 $/\$
69	68	com23 86	. . . 4 $/\$
70	69	rexlimdva 3031	. . 3
71	1, 70	sylbid 230	. 2
72	71	3imp 1256	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wrex 2913 class class class wbr 4653 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

c2 11070

cz 11377

cdvds 14983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-dvds 14984

This theorem is referenced by: gausslemma2dlem1a 25090

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