div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9291

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 9117	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8613	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8058	. . . . . 6
7	6	times2d 9281	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4072	. . 3 $/\$
10		4cn 9114	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9107	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8095	. . . . . 6
15		4p2e6 9180	. . . . . . 7
16	15	oveq2i 5955	. . . . . 6
17	14, 16	eqtrdi 2254	. . . . 5
18	17	breq1d 4054	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9113	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9135	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8908	. . . . . 6
26		peano2re 8208	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8339	. . . . . 6
29	28	rehalfcld 9284	. . . . 5
30		4pos 9133	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8666	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1250	. . . 4
35	25	recnd 8101	. . . . . 6
36		1cnd 8088	. . . . . 6
37	6, 11, 24	divcanap1d 8864	. . . . . . 7
38	10	mullidi 8075	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5962	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8100	. . . . 5
42		2t2e4 9191	. . . . . . . . 9
43	42	eqcomi 2209	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5960	. . . . . 6
46	29	recnd 8101	. . . . . . 7
47		mulass 8056	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2211	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1250	. . . . . 6
50	28	recnd 8101	. . . . . . . . 9
51		2ap0 9129	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8864	. . . . . . . 8
54	53	oveq1d 5959	. . . . . . 7
55	6, 36, 13	subdird 8487	. . . . . . 7
56	12	mullidi 8075	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5960	. . . . . . 7
59	54, 55, 58	3eqtrd 2242	. . . . . 6
60	45, 49, 59	3eqtrd 2242	. . . . 5
61	41, 60	breq12d 4057	. . . 4
62	3, 22	readdcld 8102	. . . . 5
63		2re 9106	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8103	. . . . 5
66		leaddsub 8511	. . . . . 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 2176 class class class wbr 4044 (class class class)co 5944

cc 7923

cr 7924

cc0 7925

c1 7926

caddc 7928

cmul 7930

clt 8107

cle 8108

cmin 8243 # cap 8654

cdiv 8745

c2 9087

c4 9089

c6 9091

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-po 4343 df-iso 4344 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099

This theorem is referenced by: fldiv4p1lem1div2 10448

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