div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9538

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 9364	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8858	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8302	. . . . . 6
7	6	times2d 9528	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4153	. . 3 $/\$
10		4cn 9361	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9354	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8338	. . . . . 6
15		4p2e6 9427	. . . . . . 7
16	15	oveq2i 6086	. . . . . 6
17	14, 16	eqtrdi 2287	. . . . 5
18	17	breq1d 4135	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9360	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9382	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9155	. . . . . 6
26		peano2re 8452	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8583	. . . . . 6
29	28	rehalfcld 9531	. . . . 5
30		4pos 9380	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8911	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1278	. . . 4
35	25	recnd 8344	. . . . . 6
36		1cnd 8332	. . . . . 6
37	6, 11, 24	divcanap1d 9111	. . . . . . 7
38	10	mullidi 8319	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6093	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8343	. . . . 5
42		2t2e4 9438	. . . . . . . . 9
43	42	eqcomi 2242	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6091	. . . . . 6
46	29	recnd 8344	. . . . . . 7
47		mulass 8300	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2244	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1278	. . . . . 6
50	28	recnd 8344	. . . . . . . . 9
51		2ap0 9376	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 9111	. . . . . . . 8
54	53	oveq1d 6090	. . . . . . 7
55	6, 36, 13	subdird 8732	. . . . . . 7
56	12	mullidi 8319	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6091	. . . . . . 7
59	54, 55, 58	3eqtrd 2275	. . . . . 6
60	45, 49, 59	3eqtrd 2275	. . . . 5
61	41, 60	breq12d 4138	. . . 4
62	3, 22	readdcld 8345	. . . . 5
63		2re 9353	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8346	. . . . 5
66		leaddsub 8756	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1278	. . . 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 1009

wceq 1402

wcel 2209 class class class wbr 4125 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487 # cap 8899

cdiv 8992

c2 9334

c4 9336

c6 9338

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346

This theorem is referenced by: fldiv4p1lem1div2 10718

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