div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9492

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 9318	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8814	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8260	. . . . . 6
7	6	times2d 9482	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4137	. . 3 $/\$
10		4cn 9315	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9308	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8296	. . . . . 6
15		4p2e6 9381	. . . . . . 7
16	15	oveq2i 6061	. . . . . 6
17	14, 16	eqtrdi 2281	. . . . 5
18	17	breq1d 4119	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9314	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9336	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9109	. . . . . 6
26		peano2re 8409	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8540	. . . . . 6
29	28	rehalfcld 9485	. . . . 5
30		4pos 9334	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8867	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1274	. . . 4
35	25	recnd 8302	. . . . . 6
36		1cnd 8290	. . . . . 6
37	6, 11, 24	divcanap1d 9065	. . . . . . 7
38	10	mullidi 8277	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6068	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8301	. . . . 5
42		2t2e4 9392	. . . . . . . . 9
43	42	eqcomi 2236	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6066	. . . . . 6
46	29	recnd 8302	. . . . . . 7
47		mulass 8258	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2238	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1274	. . . . . 6
50	28	recnd 8302	. . . . . . . . 9
51		2ap0 9330	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 9065	. . . . . . . 8
54	53	oveq1d 6065	. . . . . . 7
55	6, 36, 13	subdird 8688	. . . . . . 7
56	12	mullidi 8277	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6066	. . . . . . 7
59	54, 55, 58	3eqtrd 2269	. . . . . 6
60	45, 49, 59	3eqtrd 2269	. . . . 5
61	41, 60	breq12d 4122	. . . 4
62	3, 22	readdcld 8303	. . . . 5
63		2re 9307	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8304	. . . . 5
66		leaddsub 8712	. . . . . 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 2203 class class class wbr 4109 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

c1 8128

caddc 8130

cmul 8132

clt 8308

cle 8309

cmin 8444 # cap 8855

cdiv 8946

c2 9288

c4 9290

c6 9292

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 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-po 4417 df-iso 4418 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-2 9296 df-3 9297 df-4 9298 df-5 9299 df-6 9300

This theorem is referenced by: fldiv4p1lem1div2 10665

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