div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9376

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 9202	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8698	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8143	. . . . . 6
7	6	times2d 9366	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4111	. . 3 $/\$
10		4cn 9199	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9192	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8180	. . . . . 6
15		4p2e6 9265	. . . . . . 7
16	15	oveq2i 6018	. . . . . 6
17	14, 16	eqtrdi 2278	. . . . 5
18	17	breq1d 4093	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9198	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9220	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8993	. . . . . 6
26		peano2re 8293	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8424	. . . . . 6
29	28	rehalfcld 9369	. . . . 5
30		4pos 9218	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8751	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1271	. . . 4
35	25	recnd 8186	. . . . . 6
36		1cnd 8173	. . . . . 6
37	6, 11, 24	divcanap1d 8949	. . . . . . 7
38	10	mullidi 8160	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6025	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8185	. . . . 5
42		2t2e4 9276	. . . . . . . . 9
43	42	eqcomi 2233	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6023	. . . . . 6
46	29	recnd 8186	. . . . . . 7
47		mulass 8141	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2235	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1271	. . . . . 6
50	28	recnd 8186	. . . . . . . . 9
51		2ap0 9214	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8949	. . . . . . . 8
54	53	oveq1d 6022	. . . . . . 7
55	6, 36, 13	subdird 8572	. . . . . . 7
56	12	mullidi 8160	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6023	. . . . . . 7
59	54, 55, 58	3eqtrd 2266	. . . . . 6
60	45, 49, 59	3eqtrd 2266	. . . . 5
61	41, 60	breq12d 4096	. . . 4
62	3, 22	readdcld 8187	. . . . 5
63		2re 9191	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8188	. . . . 5
66		leaddsub 8596	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1271	. . . 4
69	34, 61, 68	3bitrd 214	. . 3
70	69	adantr 276	. 2 $/\$
71	20, 70	mpbird 167	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1002

wceq 1395

wcel 2200 class class class wbr 4083 (class class class)co 6007

cc 8008

cr 8009

cc0 8010

c1 8011

caddc 8013

cmul 8015

clt 8192

cle 8193

cmin 8328 # cap 8739

cdiv 8830

c2 9172

c4 9174

c6 9176

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 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128

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-rmo 2516 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 3889 df-br 4084 df-opab 4146 df-id 4384 df-po 4387 df-iso 4388 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-2 9180 df-3 9181 df-4 9182 df-5 9183 df-6 9184

This theorem is referenced by: fldiv4p1lem1div2 10537

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