div4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem div4p1lem1div2 9101

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 8929	. . . . . . 7
2	1	a1i 9	. . . . . 6
3		id 19	. . . . . 6
4	2, 3, 3	leadd2d 8429	. . . . 5
5	4	biimpa 294	. . . 4 $/\$
6		recn 7877	. . . . . 6
7	6	times2d 9091	. . . . 5
8	7	adantr 274	. . . 4 $/\$
9	5, 8	breqtrrd 4004	. . 3 $/\$
10		4cn 8926	. . . . . . . 8
11	10	a1i 9	. . . . . . 7
12		2cn 8919	. . . . . . . 8
13	12	a1i 9	. . . . . . 7
14	6, 11, 13	addassd 7912	. . . . . 6
15		4p2e6 8991	. . . . . . 7
16	15	oveq2i 5847	. . . . . 6
17	14, 16	eqtrdi 2213	. . . . 5
18	17	breq1d 3986	. . . 4
19	18	adantr 274	. . 3 $/\$
20	9, 19	mpbird 166	. 2 $/\$
21		4re 8925	. . . . . . . 8
22	21	a1i 9	. . . . . . 7
23		4ap0 8947	. . . . . . . 8 #
24	23	a1i 9	. . . . . . 7 #
25	3, 22, 24	redivclapd 8722	. . . . . 6
26		peano2re 8025	. . . . . 6
27	25, 26	syl 14	. . . . 5
28		peano2rem 8156	. . . . . 6
29	28	rehalfcld 9094	. . . . 5
30		4pos 8945	. . . . . . 7
31	21, 30	pm3.2i 270	. . . . . 6 $/\$
32	31	a1i 9	. . . . 5 $/\$
33		lemul1 8482	. . . . 5 $/\$ $/\$ $/\$
34	27, 29, 32, 33	syl3anc 1227	. . . 4
35	25	recnd 7918	. . . . . 6
36		1cnd 7906	. . . . . 6
37	6, 11, 24	divcanap1d 8678	. . . . . . 7
38	10	mulid2i 7893	. . . . . . . 8
39	38	a1i 9	. . . . . . 7
40	37, 39	oveq12d 5854	. . . . . 6
41	35, 11, 36, 40	joinlmuladdmuld 7917	. . . . 5
42		2t2e4 9002	. . . . . . . . 9
43	42	eqcomi 2168	. . . . . . . 8
44	43	a1i 9	. . . . . . 7
45	44	oveq2d 5852	. . . . . 6
46	29	recnd 7918	. . . . . . 7
47		mulass 7875	. . . . . . . 8 $/\$ $/\$
48	47	eqcomd 2170	. . . . . . 7 $/\$ $/\$
49	46, 13, 13, 48	syl3anc 1227	. . . . . 6
50	28	recnd 7918	. . . . . . . . 9
51		2ap0 8941	. . . . . . . . . 10 #
52	51	a1i 9	. . . . . . . . 9 #
53	50, 13, 52	divcanap1d 8678	. . . . . . . 8
54	53	oveq1d 5851	. . . . . . 7
55	6, 36, 13	subdird 8304	. . . . . . 7
56	12	mulid2i 7893	. . . . . . . . 9
57	56	a1i 9	. . . . . . . 8
58	57	oveq2d 5852	. . . . . . 7
59	54, 55, 58	3eqtrd 2201	. . . . . 6
60	45, 49, 59	3eqtrd 2201	. . . . 5
61	41, 60	breq12d 3989	. . . 4
62	3, 22	readdcld 7919	. . . . 5
63		2re 8918	. . . . . 6
64	63	a1i 9	. . . . 5
65	3, 64	remulcld 7920	. . . . 5
66		leaddsub 8327	. . . . . 6 $/\$ $/\$
67	66	bicomd 140	. . . . 5 $/\$ $/\$
68	62, 64, 65, 67	syl3anc 1227	. . . 4
69	34, 61, 68	3bitrd 213	. . 3
70	69	adantr 274	. 2 $/\$
71	20, 70	mpbird 166	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 967

wceq 1342

wcel 2135 class class class wbr 3976 (class class class)co 5836

cc 7742

cr 7743

cc0 7744

c1 7745

caddc 7747

cmul 7749

clt 7924

cle 7925

cmin 8060 # cap 8470

cdiv 8559

c2 8899

c4 8901

c6 8903

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911

This theorem is referenced by: fldiv4p1lem1div2 10230

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