div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9131

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 8959	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8459	. . . . 5
5	4	biimpa 294	. . . 4 $/\$
6		recn 7907	. . . . . 6
7	6	times2d 9121	. . . . 5
8	7	adantr 274	. . . 4 $/\$
9	5, 8	breqtrrd 4017	. . 3 $/\$
10		4cn 8956	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8949	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7942	. . . . . 6
15		4p2e6 9021	. . . . . . 7
16	15	oveq2i 5864	. . . . . 6
17	14, 16	eqtrdi 2219	. . . . 5
18	17	breq1d 3999	. . . 4
19	18	adantr 274	. . 3 $/\$
20	9, 19	mpbird 166	. 2 $/\$
21		4re 8955	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 8977	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8752	. . . . . 6
26		peano2re 8055	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8186	. . . . . 6
29	28	rehalfcld 9124	. . . . 5
30		4pos 8975	. . . . . . 7
31	21, 30	pm3.2i 270	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8512	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1233	. . . 4
35	25	recnd 7948	. . . . . 6
36		1cnd 7936	. . . . . 6
37	6, 11, 24	divcanap1d 8708	. . . . . . 7
38	10	mulid2i 7923	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5871	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7947	. . . . 5
42		2t2e4 9032	. . . . . . . . 9
43	42	eqcomi 2174	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5869	. . . . . 6
46	29	recnd 7948	. . . . . . 7
47		mulass 7905	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2176	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1233	. . . . . 6
50	28	recnd 7948	. . . . . . . . 9
51		2ap0 8971	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8708	. . . . . . . 8
54	53	oveq1d 5868	. . . . . . 7
55	6, 36, 13	subdird 8334	. . . . . . 7
56	12	mulid2i 7923	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5869	. . . . . . 7
59	54, 55, 58	3eqtrd 2207	. . . . . 6
60	45, 49, 59	3eqtrd 2207	. . . . 5
61	41, 60	breq12d 4002	. . . 4
62	3, 22	readdcld 7949	. . . . 5
63		2re 8948	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7950	. . . . 5
66		leaddsub 8357	. . . . . 6 $/\$ $/\$
67	66	bicomd 140	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1233	. . . 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 973

wceq 1348

wcel 2141 class class class wbr 3989 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090 # cap 8500

cdiv 8589

c2 8929

c4 8931

c6 8933

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941

This theorem is referenced by: fldiv4p1lem1div2 10261

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