div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9397

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 9223	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8719	. . . . 5
5	4	biimpa 296	. . . 4 $/\$
6		recn 8164	. . . . . 6
7	6	times2d 9387	. . . . 5
8	7	adantr 276	. . . 4 $/\$
9	5, 8	breqtrrd 4116	. . 3 $/\$
10		4cn 9220	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 9213	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 8201	. . . . . 6
15		4p2e6 9286	. . . . . . 7
16	15	oveq2i 6028	. . . . . 6
17	14, 16	eqtrdi 2280	. . . . 5
18	17	breq1d 4098	. . . 4
19	18	adantr 276	. . 3 $/\$
20	9, 19	mpbird 167	. 2 $/\$
21		4re 9219	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 9241	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 9014	. . . . . 6
26		peano2re 8314	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8445	. . . . . 6
29	28	rehalfcld 9390	. . . . 5
30		4pos 9239	. . . . . . 7
31	21, 30	pm3.2i 272	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8772	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1273	. . . 4
35	25	recnd 8207	. . . . . 6
36		1cnd 8194	. . . . . 6
37	6, 11, 24	divcanap1d 8970	. . . . . . 7
38	10	mullidi 8181	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 6035	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 8206	. . . . 5
42		2t2e4 9297	. . . . . . . . 9
43	42	eqcomi 2235	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 6033	. . . . . 6
46	29	recnd 8207	. . . . . . 7
47		mulass 8162	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2237	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1273	. . . . . 6
50	28	recnd 8207	. . . . . . . . 9
51		2ap0 9235	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8970	. . . . . . . 8
54	53	oveq1d 6032	. . . . . . 7
55	6, 36, 13	subdird 8593	. . . . . . 7
56	12	mullidi 8181	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 6033	. . . . . . 7
59	54, 55, 58	3eqtrd 2268	. . . . . 6
60	45, 49, 59	3eqtrd 2268	. . . . 5
61	41, 60	breq12d 4101	. . . 4
62	3, 22	readdcld 8208	. . . . 5
63		2re 9212	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 8209	. . . . 5
66		leaddsub 8617	. . . . . 6 $/\$ $/\$
67	66	bicomd 141	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1273	. . . 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 1004

wceq 1397

wcel 2202 class class class wbr 4088 (class class class)co 6017

cc 8029

cr 8030

cc0 8031

c1 8032

caddc 8034

cmul 8036

clt 8213

cle 8214

cmin 8349 # cap 8760

cdiv 8851

c2 9193

c4 9195

c6 9197

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205

This theorem is referenced by: fldiv4p1lem1div2 10564

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