div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9388

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 9214	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8710	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8155	. . . . . 6
7	6	times2d 9378	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4114	. . 3 $/\$
10		4cn 9211	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9204	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8192	. . . . . 6
15		4p2e6 9277	. . . . . . 7
16	15	oveq2i 6024	. . . . . 6
17	14, 16	eqtrdi 2278	. . . . 5
18	17	breq1d 4096	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9210	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9232	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9005	. . . . . 6
26		peano2re 8305	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8436	. . . . . 6
29	28	rehalfcld 9381	. . . . 5
30		4pos 9230	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8763	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1271	. . . 4
35	25	recnd 8198	. . . . . 6
36		1cnd 8185	. . . . . 6
37	6, 11, 24	divcanap1d 8961	. . . . . . 7
38	10	mullidi 8172	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6031	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8197	. . . . 5
42		2t2e4 9288	. . . . . . . . 9
43	42	eqcomi 2233	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6029	. . . . . 6
46	29	recnd 8198	. . . . . . 7
47		mulass 8153	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2235	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1271	. . . . . 6
50	28	recnd 8198	. . . . . . . . 9
51		2ap0 9226	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8961	. . . . . . . 8
54	53	oveq1d 6028	. . . . . . 7
55	6, 36, 13	subdird 8584	. . . . . . 7
56	12	mullidi 8172	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6029	. . . . . . 7
59	54, 55, 58	3eqtrd 2266	. . . . . 6
60	45, 49, 59	3eqtrd 2266	. . . . 5
61	41, 60	breq12d 4099	. . . 4
62	3, 22	readdcld 8199	. . . . 5
63		2re 9203	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8200	. . . . 5
66		leaddsub 8608	. . . . . 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 4086 (class class class)co 6013

cc 8020

cr 8021

cc0 8022

c1 8023

caddc 8025

cmul 8027

clt 8204

cle 8205

cmin 8340 # cap 8751

cdiv 8842

c2 9184

c4 9186

c6 9188

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 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140

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 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-po 4391 df-iso 4392 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196

This theorem is referenced by: fldiv4p1lem1div2 10555

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