div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9491

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 9317	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8813	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8259	. . . . . 6
7	6	times2d 9481	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4136	. . 3 $/\$
10		4cn 9314	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9307	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8295	. . . . . 6
15		4p2e6 9380	. . . . . . 7
16	15	oveq2i 6060	. . . . . 6
17	14, 16	eqtrdi 2281	. . . . 5
18	17	breq1d 4118	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9313	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9335	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9108	. . . . . 6
26		peano2re 8408	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8539	. . . . . 6
29	28	rehalfcld 9484	. . . . 5
30		4pos 9333	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8866	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1274	. . . 4
35	25	recnd 8301	. . . . . 6
36		1cnd 8289	. . . . . 6
37	6, 11, 24	divcanap1d 9064	. . . . . . 7
38	10	mullidi 8276	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6067	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8300	. . . . 5
42		2t2e4 9391	. . . . . . . . 9
43	42	eqcomi 2236	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6065	. . . . . 6
46	29	recnd 8301	. . . . . . 7
47		mulass 8257	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2238	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1274	. . . . . 6
50	28	recnd 8301	. . . . . . . . 9
51		2ap0 9329	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 9064	. . . . . . . 8
54	53	oveq1d 6064	. . . . . . 7
55	6, 36, 13	subdird 8687	. . . . . . 7
56	12	mullidi 8276	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6065	. . . . . . 7
59	54, 55, 58	3eqtrd 2269	. . . . . 6
60	45, 49, 59	3eqtrd 2269	. . . . 5
61	41, 60	breq12d 4121	. . . 4
62	3, 22	readdcld 8302	. . . . 5
63		2re 9306	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8303	. . . . 5
66		leaddsub 8711	. . . . . 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 4108 (class class class)co 6049

cc 8124

cr 8125

cc0 8126

c1 8127

caddc 8129

cmul 8131

clt 8307

cle 8308

cmin 8443 # cap 8854

cdiv 8945

c2 9287

c4 9289

c6 9291

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 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-cnex 8217 ax-resscn 8218 ax-1cn 8219 ax-1re 8220 ax-icn 8221 ax-addcl 8222 ax-addrcl 8223 ax-mulcl 8224 ax-mulrcl 8225 ax-addcom 8226 ax-mulcom 8227 ax-addass 8228 ax-mulass 8229 ax-distr 8230 ax-i2m1 8231 ax-0lt1 8232 ax-1rid 8233 ax-0id 8234 ax-rnegex 8235 ax-precex 8236 ax-cnre 8237 ax-pre-ltirr 8238 ax-pre-ltwlin 8239 ax-pre-lttrn 8240 ax-pre-apti 8241 ax-pre-ltadd 8242 ax-pre-mulgt0 8243 ax-pre-mulext 8244

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 2814 df-sbc 3042 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-br 4109 df-opab 4171 df-id 4413 df-po 4416 df-iso 4417 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-iota 5311 df-fun 5353 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-pnf 8309 df-mnf 8310 df-xr 8311 df-ltxr 8312 df-le 8313 df-sub 8445 df-neg 8446 df-reap 8848 df-ap 8855 df-div 8946 df-2 9295 df-3 9296 df-4 9297 df-5 9298 df-6 9299

This theorem is referenced by: fldiv4p1lem1div2 10664

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