div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9236

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 9063	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8559	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8005	. . . . . 6
7	6	times2d 9226	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4057	. . 3 $/\$
10		4cn 9060	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9053	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8042	. . . . . 6
15		4p2e6 9125	. . . . . . 7
16	15	oveq2i 5929	. . . . . 6
17	14, 16	eqtrdi 2242	. . . . 5
18	17	breq1d 4039	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9059	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9081	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8854	. . . . . 6
26		peano2re 8155	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8286	. . . . . 6
29	28	rehalfcld 9229	. . . . 5
30		4pos 9079	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8612	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1249	. . . 4
35	25	recnd 8048	. . . . . 6
36		1cnd 8035	. . . . . 6
37	6, 11, 24	divcanap1d 8810	. . . . . . 7
38	10	mullidi 8022	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5936	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8047	. . . . 5
42		2t2e4 9136	. . . . . . . . 9
43	42	eqcomi 2197	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5934	. . . . . 6
46	29	recnd 8048	. . . . . . 7
47		mulass 8003	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2199	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1249	. . . . . 6
50	28	recnd 8048	. . . . . . . . 9
51		2ap0 9075	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8810	. . . . . . . 8
54	53	oveq1d 5933	. . . . . . 7
55	6, 36, 13	subdird 8434	. . . . . . 7
56	12	mullidi 8022	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5934	. . . . . . 7
59	54, 55, 58	3eqtrd 2230	. . . . . 6
60	45, 49, 59	3eqtrd 2230	. . . . 5
61	41, 60	breq12d 4042	. . . 4
62	3, 22	readdcld 8049	. . . . 5
63		2re 9052	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8050	. . . . 5
66		leaddsub 8457	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1249	. . . 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 980

wceq 1364

wcel 2164 class class class wbr 4029 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190 # cap 8600

cdiv 8691

c2 9033

c4 9035

c6 9037

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-2 9041 df-3 9042 df-4 9043 df-5 9044 df-6 9045

This theorem is referenced by: fldiv4p1lem1div2 10374

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