div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9440

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 9266	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8762	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8208	. . . . . 6
7	6	times2d 9430	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4121	. . 3 $/\$
10		4cn 9263	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9256	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8244	. . . . . 6
15		4p2e6 9329	. . . . . . 7
16	15	oveq2i 6039	. . . . . 6
17	14, 16	eqtrdi 2280	. . . . 5
18	17	breq1d 4103	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9262	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9284	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9057	. . . . . 6
26		peano2re 8357	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8488	. . . . . 6
29	28	rehalfcld 9433	. . . . 5
30		4pos 9282	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8815	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1274	. . . 4
35	25	recnd 8250	. . . . . 6
36		1cnd 8238	. . . . . 6
37	6, 11, 24	divcanap1d 9013	. . . . . . 7
38	10	mullidi 8225	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6046	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8249	. . . . 5
42		2t2e4 9340	. . . . . . . . 9
43	42	eqcomi 2235	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6044	. . . . . 6
46	29	recnd 8250	. . . . . . 7
47		mulass 8206	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2237	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1274	. . . . . 6
50	28	recnd 8250	. . . . . . . . 9
51		2ap0 9278	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 9013	. . . . . . . 8
54	53	oveq1d 6043	. . . . . . 7
55	6, 36, 13	subdird 8636	. . . . . . 7
56	12	mullidi 8225	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6044	. . . . . . 7
59	54, 55, 58	3eqtrd 2268	. . . . . 6
60	45, 49, 59	3eqtrd 2268	. . . . 5
61	41, 60	breq12d 4106	. . . 4
62	3, 22	readdcld 8251	. . . . 5
63		2re 9255	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8252	. . . . 5
66		leaddsub 8660	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1274	. . . 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 1005

wceq 1398

wcel 2202 class class class wbr 4093 (class class class)co 6028

cc 8073

cr 8074

cc0 8075

c1 8076

caddc 8078

cmul 8080

clt 8256

cle 8257

cmin 8392 # cap 8803

cdiv 8894

c2 9236

c4 9238

c6 9240

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-id 4396 df-po 4399 df-iso 4400 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248

This theorem is referenced by: fldiv4p1lem1div2 10611

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