div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9326

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 9152	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8648	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8093	. . . . . 6
7	6	times2d 9316	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4087	. . 3 $/\$
10		4cn 9149	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9142	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8130	. . . . . 6
15		4p2e6 9215	. . . . . . 7
16	15	oveq2i 5978	. . . . . 6
17	14, 16	eqtrdi 2256	. . . . 5
18	17	breq1d 4069	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9148	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9170	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8943	. . . . . 6
26		peano2re 8243	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8374	. . . . . 6
29	28	rehalfcld 9319	. . . . 5
30		4pos 9168	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8701	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1250	. . . 4
35	25	recnd 8136	. . . . . 6
36		1cnd 8123	. . . . . 6
37	6, 11, 24	divcanap1d 8899	. . . . . . 7
38	10	mullidi 8110	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5985	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8135	. . . . 5
42		2t2e4 9226	. . . . . . . . 9
43	42	eqcomi 2211	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5983	. . . . . 6
46	29	recnd 8136	. . . . . . 7
47		mulass 8091	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2213	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1250	. . . . . 6
50	28	recnd 8136	. . . . . . . . 9
51		2ap0 9164	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8899	. . . . . . . 8
54	53	oveq1d 5982	. . . . . . 7
55	6, 36, 13	subdird 8522	. . . . . . 7
56	12	mullidi 8110	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5983	. . . . . . 7
59	54, 55, 58	3eqtrd 2244	. . . . . 6
60	45, 49, 59	3eqtrd 2244	. . . . 5
61	41, 60	breq12d 4072	. . . 4
62	3, 22	readdcld 8137	. . . . 5
63		2re 9141	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8138	. . . . 5
66		leaddsub 8546	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1250	. . . 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 981

wceq 1373

wcel 2178 class class class wbr 4059 (class class class)co 5967

cc 7958

cr 7959

cc0 7960

c1 7961

caddc 7963

cmul 7965

clt 8142

cle 8143

cmin 8278 # cap 8689

cdiv 8780

c2 9122

c4 9124

c6 9126

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-id 4358 df-po 4361 df-iso 4362 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-iota 5251 df-fun 5292 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-2 9130 df-3 9131 df-4 9132 df-5 9133 df-6 9134

This theorem is referenced by: fldiv4p1lem1div2 10485

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