div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9361

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 9187	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8683	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8128	. . . . . 6
7	6	times2d 9351	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4110	. . 3 $/\$
10		4cn 9184	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9177	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8165	. . . . . 6
15		4p2e6 9250	. . . . . . 7
16	15	oveq2i 6011	. . . . . 6
17	14, 16	eqtrdi 2278	. . . . 5
18	17	breq1d 4092	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9183	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9205	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8978	. . . . . 6
26		peano2re 8278	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8409	. . . . . 6
29	28	rehalfcld 9354	. . . . 5
30		4pos 9203	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8736	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1271	. . . 4
35	25	recnd 8171	. . . . . 6
36		1cnd 8158	. . . . . 6
37	6, 11, 24	divcanap1d 8934	. . . . . . 7
38	10	mullidi 8145	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6018	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8170	. . . . 5
42		2t2e4 9261	. . . . . . . . 9
43	42	eqcomi 2233	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6016	. . . . . 6
46	29	recnd 8171	. . . . . . 7
47		mulass 8126	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2235	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1271	. . . . . 6
50	28	recnd 8171	. . . . . . . . 9
51		2ap0 9199	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8934	. . . . . . . 8
54	53	oveq1d 6015	. . . . . . 7
55	6, 36, 13	subdird 8557	. . . . . . 7
56	12	mullidi 8145	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6016	. . . . . . 7
59	54, 55, 58	3eqtrd 2266	. . . . . 6
60	45, 49, 59	3eqtrd 2266	. . . . 5
61	41, 60	breq12d 4095	. . . 4
62	3, 22	readdcld 8172	. . . . 5
63		2re 9176	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8173	. . . . 5
66		leaddsub 8581	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1271	. . . 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 1002

wceq 1395

wcel 2200 class class class wbr 4082 (class class class)co 6000

cc 7993

cr 7994

cc0 7995

c1 7996

caddc 7998

cmul 8000

clt 8177

cle 8178

cmin 8313 # cap 8724

cdiv 8815

c2 9157

c4 9159

c6 9161

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-br 4083 df-opab 4145 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-iota 5277 df-fun 5319 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-2 9165 df-3 9166 df-4 9167 df-5 9168 df-6 9169

This theorem is referenced by: fldiv4p1lem1div2 10520

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