div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9110

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 8938	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8438	. . . . 5
5	4	biimpa 294	. . . 4 $/\$
6		recn 7886	. . . . . 6
7	6	times2d 9100	. . . . 5
8	7	adantr 274	. . . 4 $/\$
9	5, 8	breqtrrd 4010	. . 3 $/\$
10		4cn 8935	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8928	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7921	. . . . . 6
15		4p2e6 9000	. . . . . . 7
16	15	oveq2i 5853	. . . . . 6
17	14, 16	eqtrdi 2215	. . . . 5
18	17	breq1d 3992	. . . 4
19	18	adantr 274	. . 3 $/\$
20	9, 19	mpbird 166	. 2 $/\$
21		4re 8934	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 8956	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8731	. . . . . 6
26		peano2re 8034	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8165	. . . . . 6
29	28	rehalfcld 9103	. . . . 5
30		4pos 8954	. . . . . . 7
31	21, 30	pm3.2i 270	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8491	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1228	. . . 4
35	25	recnd 7927	. . . . . 6
36		1cnd 7915	. . . . . 6
37	6, 11, 24	divcanap1d 8687	. . . . . . 7
38	10	mulid2i 7902	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5860	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7926	. . . . 5
42		2t2e4 9011	. . . . . . . . 9
43	42	eqcomi 2169	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5858	. . . . . 6
46	29	recnd 7927	. . . . . . 7
47		mulass 7884	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2171	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1228	. . . . . 6
50	28	recnd 7927	. . . . . . . . 9
51		2ap0 8950	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8687	. . . . . . . 8
54	53	oveq1d 5857	. . . . . . 7
55	6, 36, 13	subdird 8313	. . . . . . 7
56	12	mulid2i 7902	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5858	. . . . . . 7
59	54, 55, 58	3eqtrd 2202	. . . . . 6
60	45, 49, 59	3eqtrd 2202	. . . . 5
61	41, 60	breq12d 3995	. . . 4
62	3, 22	readdcld 7928	. . . . 5
63		2re 8927	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7929	. . . . 5
66		leaddsub 8336	. . . . . 6 $/\$ $/\$
67	66	bicomd 140	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1228	. . . 4
69	34, 61, 68	3bitrd 213	. . 3
70	69	adantr 274	. 2 $/\$
71	20, 70	mpbird 166	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 968

wceq 1343

wcel 2136 class class class wbr 3982 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

c1 7754

caddc 7756

cmul 7758

clt 7933

cle 7934

cmin 8069 # cap 8479

cdiv 8568

c2 8908

c4 8910

c6 8912

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920

This theorem is referenced by: fldiv4p1lem1div2 10240

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