div4p1lem1div2 - Metamath Proof Explorer

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Theorem div4p1lem1div2 11287

Description: An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)

**Assertion**
Ref	Expression
div4p1lem1div2	$/\$

**Proof of Theorem div4p1lem1div2**
Step	Hyp	Ref	Expression
1		6re 11101	. . . . . . 7
2	1	a1i 11	. . . . . 6
3		id 22	. . . . . 6
4	2, 3, 3	leadd2d 10622	. . . . 5
5	4	biimpa 501	. . . 4 $/\$
6		recn 10026	. . . . . 6
7	6	times2d 11276	. . . . 5
8	7	adantr 481	. . . 4 $/\$
9	5, 8	breqtrrd 4681	. . 3 $/\$
10		4cn 11098	. . . . . . . 8
11	10	a1i 11	. . . . . . 7
12		2cn 11091	. . . . . . . 8
13	12	a1i 11	. . . . . . 7
14	6, 11, 13	addassd 10062	. . . . . 6
15		4p2e6 11162	. . . . . . 7
16	15	oveq2i 6661	. . . . . 6
17	14, 16	syl6eq 2672	. . . . 5
18	17	breq1d 4663	. . . 4
19	18	adantr 481	. . 3 $/\$
20	9, 19	mpbird 247	. 2 $/\$
21		4re 11097	. . . . . . . 8
22	21	a1i 11	. . . . . . 7
23		4ne0 11117	. . . . . . . 8
24	23	a1i 11	. . . . . . 7
25	3, 22, 24	redivcld 10853	. . . . . 6
26		peano2re 10209	. . . . . 6
27	25, 26	syl 17	. . . . 5
28		peano2rem 10348	. . . . . 6
29	28	rehalfcld 11279	. . . . 5
30		4pos 11116	. . . . . . 7
31	21, 30	pm3.2i 471	. . . . . 6 $/\$
32	31	a1i 11	. . . . 5 $/\$
33		lemul1 10875	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1326	. . . 4
35	25	recnd 10068	. . . . . 6
36		1cnd 10056	. . . . . 6
37	6, 11, 24	divcan1d 10802	. . . . . . 7
38	10	mulid2i 10043	. . . . . . . 8
39	38	a1i 11	. . . . . . 7
40	37, 39	oveq12d 6668	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 10067	. . . . 5
42		2t2e4 11177	. . . . . . . . 9
43	42	eqcomi 2631	. . . . . . . 8
44	43	a1i 11	. . . . . . 7
45	44	oveq2d 6666	. . . . . 6
46	29	recnd 10068	. . . . . . 7
47		mulass 10024	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2628	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1326	. . . . . 6
50	28	recnd 10068	. . . . . . . . 9
51		2ne0 11113	. . . . . . . . . 10
52	51	a1i 11	. . . . . . . . 9
53	50, 13, 52	divcan1d 10802	. . . . . . . 8
54	53	oveq1d 6665	. . . . . . 7
55	6, 36, 13	subdird 10487	. . . . . . 7
56	12	mulid2i 10043	. . . . . . . . 9
57	56	a1i 11	. . . . . . . 8
58	57	oveq2d 6666	. . . . . . 7
59	54, 55, 58	3eqtrd 2660	. . . . . 6
60	45, 49, 59	3eqtrd 2660	. . . . 5
61	41, 60	breq12d 4666	. . . 4
62	3, 22	readdcld 10069	. . . . 5
63		2re 11090	. . . . . 6
64	63	a1i 11	. . . . 5
65	3, 64	remulcld 10070	. . . . 5
66		leaddsub 10504	. . . . . 6 $/\$ $/\$
67	66	bicomd 213	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1326	. . . 4
69	34, 61, 68	3bitrd 294	. . 3
70	69	adantr 481	. 2 $/\$
71	20, 70	mpbird 247	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990

wne 2794 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

c4 11072

c6 11074

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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-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-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-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083

This theorem is referenced by: fldiv4p1lem1div2 12636

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