div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9168

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 8996	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8493	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 7941	. . . . . 6
7	6	times2d 9158	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4030	. . 3 $/\$
10		4cn 8993	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8986	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7976	. . . . . 6
15		4p2e6 9058	. . . . . . 7
16	15	oveq2i 5883	. . . . . 6
17	14, 16	eqtrdi 2226	. . . . 5
18	17	breq1d 4012	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 8992	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9014	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8788	. . . . . 6
26		peano2re 8089	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8220	. . . . . 6
29	28	rehalfcld 9161	. . . . 5
30		4pos 9012	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8546	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1238	. . . 4
35	25	recnd 7982	. . . . . 6
36		1cnd 7970	. . . . . 6
37	6, 11, 24	divcanap1d 8744	. . . . . . 7
38	10	mulid2i 7957	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5890	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7981	. . . . 5
42		2t2e4 9069	. . . . . . . . 9
43	42	eqcomi 2181	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5888	. . . . . 6
46	29	recnd 7982	. . . . . . 7
47		mulass 7939	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2183	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1238	. . . . . 6
50	28	recnd 7982	. . . . . . . . 9
51		2ap0 9008	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8744	. . . . . . . 8
54	53	oveq1d 5887	. . . . . . 7
55	6, 36, 13	subdird 8368	. . . . . . 7
56	12	mulid2i 7957	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5888	. . . . . . 7
59	54, 55, 58	3eqtrd 2214	. . . . . 6
60	45, 49, 59	3eqtrd 2214	. . . . 5
61	41, 60	breq12d 4015	. . . 4
62	3, 22	readdcld 7983	. . . . 5
63		2re 8985	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7984	. . . . 5
66		leaddsub 8391	. . . . . 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 4002 (class class class)co 5872

cc 7806

cr 7807

cc0 7808

c1 7809

caddc 7811

cmul 7813

clt 7988

cle 7989

cmin 8124 # cap 8534

cdiv 8625

c2 8966

c4 8968

c6 8970

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 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926

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 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-po 4295 df-iso 4296 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-2 8974 df-3 8975 df-4 8976 df-5 8977 df-6 8978

This theorem is referenced by: fldiv4p1lem1div2 10300

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