div4p1lem1div2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > div4p1lem1div2		Unicode version

Theorem div4p1lem1div2 9262

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 9088	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8584	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8029	. . . . . 6
7	6	times2d 9252	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4062	. . 3 $/\$
10		4cn 9085	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9078	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8066	. . . . . 6
15		4p2e6 9151	. . . . . . 7
16	15	oveq2i 5936	. . . . . 6
17	14, 16	eqtrdi 2245	. . . . 5
18	17	breq1d 4044	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9084	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9106	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8879	. . . . . 6
26		peano2re 8179	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8310	. . . . . 6
29	28	rehalfcld 9255	. . . . 5
30		4pos 9104	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8637	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1249	. . . 4
35	25	recnd 8072	. . . . . 6
36		1cnd 8059	. . . . . 6
37	6, 11, 24	divcanap1d 8835	. . . . . . 7
38	10	mullidi 8046	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5943	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8071	. . . . 5
42		2t2e4 9162	. . . . . . . . 9
43	42	eqcomi 2200	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5941	. . . . . 6
46	29	recnd 8072	. . . . . . 7
47		mulass 8027	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2202	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1249	. . . . . 6
50	28	recnd 8072	. . . . . . . . 9
51		2ap0 9100	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8835	. . . . . . . 8
54	53	oveq1d 5940	. . . . . . 7
55	6, 36, 13	subdird 8458	. . . . . . 7
56	12	mullidi 8046	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5941	. . . . . . 7
59	54, 55, 58	3eqtrd 2233	. . . . . 6
60	45, 49, 59	3eqtrd 2233	. . . . 5
61	41, 60	breq12d 4047	. . . 4
62	3, 22	readdcld 8073	. . . . 5
63		2re 9077	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8074	. . . . 5
66		leaddsub 8482	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1249	. . . 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 980

wceq 1364

wcel 2167 class class class wbr 4034 (class class class)co 5925

cc 7894

cr 7895

cc0 7896

c1 7897

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214 # cap 8625

cdiv 8716

c2 9058

c4 9060

c6 9062

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070

This theorem is referenced by: fldiv4p1lem1div2 10412

W3C validator