div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 8284

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 8120	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 7640	. . . . 5
5	4	biimpa 290	. . . 4 $/\$
6		recn 7106	. . . . . 6
7	6	times2d 8274	. . . . 5
8	7	adantr 270	. . . 4 $/\$
9	5, 8	breqtrrd 3811	. . 3 $/\$
10		4cn 8117	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8110	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7141	. . . . . 6
15		4p2e6 8175	. . . . . . 7
16	15	oveq2i 5543	. . . . . 6
17	14, 16	syl6eq 2129	. . . . 5
18	17	breq1d 3795	. . . 4
19	18	adantr 270	. . 3 $/\$
20	9, 19	mpbird 165	. 2 $/\$
21		4re 8116	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 8138	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 7920	. . . . . 6
26		peano2re 7244	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 7375	. . . . . 6
29	28	rehalfcld 8277	. . . . 5
30		4pos 8136	. . . . . . 7
31	21, 30	pm3.2i 266	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 7693	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1169	. . . 4
35	25	recnd 7147	. . . . . 6
36		1cnd 7135	. . . . . 6
37	6, 11, 24	divcanap1d 7878	. . . . . . 7
38	10	mulid2i 7122	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5550	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7146	. . . . 5
42		2t2e4 8186	. . . . . . . . 9
43	42	eqcomi 2085	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5548	. . . . . 6
46	29	recnd 7147	. . . . . . 7
47		mulass 7104	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2086	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1169	. . . . . 6
50	28	recnd 7147	. . . . . . . . 9
51		2ap0 8132	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 7878	. . . . . . . 8
54	53	oveq1d 5547	. . . . . . 7
55	6, 36, 13	subdird 7519	. . . . . . 7
56	12	mulid2i 7122	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5548	. . . . . . 7
59	54, 55, 58	3eqtrd 2117	. . . . . 6
60	45, 49, 59	3eqtrd 2117	. . . . 5
61	41, 60	breq12d 3798	. . . 4
62	3, 22	readdcld 7148	. . . . 5
63		2re 8109	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7149	. . . . 5
66		leaddsub 7542	. . . . . 6 $/\$ $/\$
67	66	bicomd 139	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1169	. . . 4
69	34, 61, 68	3bitrd 212	. . 3
70	69	adantr 270	. 2 $/\$
71	20, 70	mpbird 165	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102

wb 103 $/\$ w3a 919

wceq 1284

wcel 1433 class class class wbr 3785 (class class class)co 5532

cc 6979

cr 6980

cc0 6981

c1 6982

caddc 6984

cmul 6986

clt 7153

cle 7154

cmin 7279 # cap 7681

cdiv 7760

c2 8089

c4 8091

c6 8093

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102

This theorem is referenced by: fldiv4p1lem1div2 9307

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