div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9174

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 9002	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8499	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 7946	. . . . . 6
7	6	times2d 9164	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4033	. . 3 $/\$
10		4cn 8999	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8992	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7982	. . . . . 6
15		4p2e6 9064	. . . . . . 7
16	15	oveq2i 5888	. . . . . 6
17	14, 16	eqtrdi 2226	. . . . 5
18	17	breq1d 4015	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 8998	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9020	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8794	. . . . . 6
26		peano2re 8095	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8226	. . . . . 6
29	28	rehalfcld 9167	. . . . 5
30		4pos 9018	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8552	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1238	. . . 4
35	25	recnd 7988	. . . . . 6
36		1cnd 7975	. . . . . 6
37	6, 11, 24	divcanap1d 8750	. . . . . . 7
38	10	mullidi 7962	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5895	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7987	. . . . 5
42		2t2e4 9075	. . . . . . . . 9
43	42	eqcomi 2181	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5893	. . . . . 6
46	29	recnd 7988	. . . . . . 7
47		mulass 7944	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2183	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1238	. . . . . 6
50	28	recnd 7988	. . . . . . . . 9
51		2ap0 9014	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8750	. . . . . . . 8
54	53	oveq1d 5892	. . . . . . 7
55	6, 36, 13	subdird 8374	. . . . . . 7
56	12	mullidi 7962	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5893	. . . . . . 7
59	54, 55, 58	3eqtrd 2214	. . . . . 6
60	45, 49, 59	3eqtrd 2214	. . . . 5
61	41, 60	breq12d 4018	. . . 4
62	3, 22	readdcld 7989	. . . . 5
63		2re 8991	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7990	. . . . 5
66		leaddsub 8397	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1238	. . . 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 978

wceq 1353

wcel 2148 class class class wbr 4005 (class class class)co 5877

cc 7811

cr 7812

cc0 7813

c1 7814

caddc 7816

cmul 7818

clt 7994

cle 7995

cmin 8130 # cap 8540

cdiv 8631

c2 8972

c4 8974

c6 8976

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984

This theorem is referenced by: fldiv4p1lem1div2 10307

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