div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9245

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 9071	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8567	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8012	. . . . . 6
7	6	times2d 9235	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4061	. . 3 $/\$
10		4cn 9068	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9061	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8049	. . . . . 6
15		4p2e6 9134	. . . . . . 7
16	15	oveq2i 5933	. . . . . 6
17	14, 16	eqtrdi 2245	. . . . 5
18	17	breq1d 4043	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9067	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9089	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8862	. . . . . 6
26		peano2re 8162	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8293	. . . . . 6
29	28	rehalfcld 9238	. . . . 5
30		4pos 9087	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8620	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1249	. . . 4
35	25	recnd 8055	. . . . . 6
36		1cnd 8042	. . . . . 6
37	6, 11, 24	divcanap1d 8818	. . . . . . 7
38	10	mullidi 8029	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5940	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8054	. . . . 5
42		2t2e4 9145	. . . . . . . . 9
43	42	eqcomi 2200	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5938	. . . . . 6
46	29	recnd 8055	. . . . . . 7
47		mulass 8010	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2202	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1249	. . . . . 6
50	28	recnd 8055	. . . . . . . . 9
51		2ap0 9083	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8818	. . . . . . . 8
54	53	oveq1d 5937	. . . . . . 7
55	6, 36, 13	subdird 8441	. . . . . . 7
56	12	mullidi 8029	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5938	. . . . . . 7
59	54, 55, 58	3eqtrd 2233	. . . . . 6
60	45, 49, 59	3eqtrd 2233	. . . . 5
61	41, 60	breq12d 4046	. . . 4
62	3, 22	readdcld 8056	. . . . 5
63		2re 9060	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8057	. . . . 5
66		leaddsub 8465	. . . . . 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 2167 class class class wbr 4033 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

c1 7880

caddc 7882

cmul 7884

clt 8061

cle 8062

cmin 8197 # cap 8608

cdiv 8699

c2 9041

c4 9043

c6 9045

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-6 9053

This theorem is referenced by: fldiv4p1lem1div2 10395

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