div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9559

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 9385	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8868	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8312	. . . . . 6
7	6	times2d 9549	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4158	. . 3 $/\$
10		4cn 9382	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9375	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8348	. . . . . 6
15		4p2e6 9448	. . . . . . 7
16	15	oveq2i 6096	. . . . . 6
17	14, 16	eqtrdi 2287	. . . . 5
18	17	breq1d 4140	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9381	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9403	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9165	. . . . . 6
26		peano2re 8462	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8593	. . . . . 6
29	28	rehalfcld 9552	. . . . 5
30		4pos 9401	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8921	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1278	. . . 4
35	25	recnd 8354	. . . . . 6
36		1cnd 8342	. . . . . 6
37	6, 11, 24	divcanap1d 9121	. . . . . . 7
38	10	mullidi 8329	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6103	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8353	. . . . 5
42		2t2e4 9459	. . . . . . . . 9
43	42	eqcomi 2242	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6101	. . . . . 6
46	29	recnd 8354	. . . . . . 7
47		mulass 8310	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2244	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1278	. . . . . 6
50	28	recnd 8354	. . . . . . . . 9
51		2ap0 9397	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 9121	. . . . . . . 8
54	53	oveq1d 6100	. . . . . . 7
55	6, 36, 13	subdird 8742	. . . . . . 7
56	12	mullidi 8329	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6101	. . . . . . 7
59	54, 55, 58	3eqtrd 2275	. . . . . 6
60	45, 49, 59	3eqtrd 2275	. . . . 5
61	41, 60	breq12d 4143	. . . 4
62	3, 22	readdcld 8355	. . . . 5
63		2re 9374	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8356	. . . . 5
66		leaddsub 8766	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1278	. . . 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

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209 class class class wbr 4130 (class class class)co 6085

cc 8177

cr 8178

cc0 8179

c1 8180

caddc 8182

cmul 8184

clt 8360

cle 8361

cmin 8497 # cap 8909

cdiv 9002

c2 9355

c4 9357

c6 9359

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-po 4441 df-iso 4442 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-2 9363 df-3 9364 df-4 9365 df-5 9366 df-6 9367

This theorem is used by: fldiv4p1lem1div2 10740

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