div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 8825

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 8659	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8168	. . . . 5
5	4	biimpa 292	. . . 4 $/\$
6		recn 7625	. . . . . 6
7	6	times2d 8815	. . . . 5
8	7	adantr 272	. . . 4 $/\$
9	5, 8	breqtrrd 3901	. . 3 $/\$
10		4cn 8656	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8649	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7660	. . . . . 6
15		4p2e6 8715	. . . . . . 7
16	15	oveq2i 5717	. . . . . 6
17	14, 16	syl6eq 2148	. . . . 5
18	17	breq1d 3885	. . . 4
19	18	adantr 272	. . 3 $/\$
20	9, 19	mpbird 166	. 2 $/\$
21		4re 8655	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 8677	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8455	. . . . . 6
26		peano2re 7769	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 7900	. . . . . 6
29	28	rehalfcld 8818	. . . . 5
30		4pos 8675	. . . . . . 7
31	21, 30	pm3.2i 268	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8221	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1184	. . . 4
35	25	recnd 7666	. . . . . 6
36		1cnd 7654	. . . . . 6
37	6, 11, 24	divcanap1d 8412	. . . . . . 7
38	10	mulid2i 7641	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5724	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7665	. . . . 5
42		2t2e4 8726	. . . . . . . . 9
43	42	eqcomi 2104	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5722	. . . . . 6
46	29	recnd 7666	. . . . . . 7
47		mulass 7623	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2105	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1184	. . . . . 6
50	28	recnd 7666	. . . . . . . . 9
51		2ap0 8671	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8412	. . . . . . . 8
54	53	oveq1d 5721	. . . . . . 7
55	6, 36, 13	subdird 8044	. . . . . . 7
56	12	mulid2i 7641	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5722	. . . . . . 7
59	54, 55, 58	3eqtrd 2136	. . . . . 6
60	45, 49, 59	3eqtrd 2136	. . . . 5
61	41, 60	breq12d 3888	. . . 4
62	3, 22	readdcld 7667	. . . . 5
63		2re 8648	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7668	. . . . 5
66		leaddsub 8067	. . . . . 6 $/\$ $/\$
67	66	bicomd 140	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1184	. . . 4
69	34, 61, 68	3bitrd 213	. . 3
70	69	adantr 272	. 2 $/\$
71	20, 70	mpbird 166	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 930

wceq 1299

wcel 1448 class class class wbr 3875 (class class class)co 5706

cc 7498

cr 7499

cc0 7500

c1 7501

caddc 7503

cmul 7505

clt 7672

cle 7673

cmin 7804 # cap 8209

cdiv 8293

c2 8629

c4 8631

c6 8633

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641

This theorem is referenced by: fldiv4p1lem1div2 9919

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