Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > div4p1lem1div2 | Unicode version |
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.) |
Ref | Expression |
---|---|
div4p1lem1div2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6re 8769 | . . . . . . 7 | |
2 | 1 | a1i 9 | . . . . . 6 |
3 | id 19 | . . . . . 6 | |
4 | 2, 3, 3 | leadd2d 8270 | . . . . 5 |
5 | 4 | biimpa 294 | . . . 4 |
6 | recn 7721 | . . . . . 6 | |
7 | 6 | times2d 8931 | . . . . 5 |
8 | 7 | adantr 274 | . . . 4 |
9 | 5, 8 | breqtrrd 3926 | . . 3 |
10 | 4cn 8766 | . . . . . . . 8 | |
11 | 10 | a1i 9 | . . . . . . 7 |
12 | 2cn 8759 | . . . . . . . 8 | |
13 | 12 | a1i 9 | . . . . . . 7 |
14 | 6, 11, 13 | addassd 7756 | . . . . . 6 |
15 | 4p2e6 8831 | . . . . . . 7 | |
16 | 15 | oveq2i 5753 | . . . . . 6 |
17 | 14, 16 | syl6eq 2166 | . . . . 5 |
18 | 17 | breq1d 3909 | . . . 4 |
19 | 18 | adantr 274 | . . 3 |
20 | 9, 19 | mpbird 166 | . 2 |
21 | 4re 8765 | . . . . . . . 8 | |
22 | 21 | a1i 9 | . . . . . . 7 |
23 | 4ap0 8787 | . . . . . . . 8 # | |
24 | 23 | a1i 9 | . . . . . . 7 # |
25 | 3, 22, 24 | redivclapd 8562 | . . . . . 6 |
26 | peano2re 7866 | . . . . . 6 | |
27 | 25, 26 | syl 14 | . . . . 5 |
28 | peano2rem 7997 | . . . . . 6 | |
29 | 28 | rehalfcld 8934 | . . . . 5 |
30 | 4pos 8785 | . . . . . . 7 | |
31 | 21, 30 | pm3.2i 270 | . . . . . 6 |
32 | 31 | a1i 9 | . . . . 5 |
33 | lemul1 8323 | . . . . 5 | |
34 | 27, 29, 32, 33 | syl3anc 1201 | . . . 4 |
35 | 25 | recnd 7762 | . . . . . 6 |
36 | 1cnd 7750 | . . . . . 6 | |
37 | 6, 11, 24 | divcanap1d 8519 | . . . . . . 7 |
38 | 10 | mulid2i 7737 | . . . . . . . 8 |
39 | 38 | a1i 9 | . . . . . . 7 |
40 | 37, 39 | oveq12d 5760 | . . . . . 6 |
41 | 35, 11, 36, 40 | joinlmuladdmuld 7761 | . . . . 5 |
42 | 2t2e4 8842 | . . . . . . . . 9 | |
43 | 42 | eqcomi 2121 | . . . . . . . 8 |
44 | 43 | a1i 9 | . . . . . . 7 |
45 | 44 | oveq2d 5758 | . . . . . 6 |
46 | 29 | recnd 7762 | . . . . . . 7 |
47 | mulass 7719 | . . . . . . . 8 | |
48 | 47 | eqcomd 2123 | . . . . . . 7 |
49 | 46, 13, 13, 48 | syl3anc 1201 | . . . . . 6 |
50 | 28 | recnd 7762 | . . . . . . . . 9 |
51 | 2ap0 8781 | . . . . . . . . . 10 # | |
52 | 51 | a1i 9 | . . . . . . . . 9 # |
53 | 50, 13, 52 | divcanap1d 8519 | . . . . . . . 8 |
54 | 53 | oveq1d 5757 | . . . . . . 7 |
55 | 6, 36, 13 | subdird 8145 | . . . . . . 7 |
56 | 12 | mulid2i 7737 | . . . . . . . . 9 |
57 | 56 | a1i 9 | . . . . . . . 8 |
58 | 57 | oveq2d 5758 | . . . . . . 7 |
59 | 54, 55, 58 | 3eqtrd 2154 | . . . . . 6 |
60 | 45, 49, 59 | 3eqtrd 2154 | . . . . 5 |
61 | 41, 60 | breq12d 3912 | . . . 4 |
62 | 3, 22 | readdcld 7763 | . . . . 5 |
63 | 2re 8758 | . . . . . 6 | |
64 | 63 | a1i 9 | . . . . 5 |
65 | 3, 64 | remulcld 7764 | . . . . 5 |
66 | leaddsub 8168 | . . . . . 6 | |
67 | 66 | bicomd 140 | . . . . 5 |
68 | 62, 64, 65, 67 | syl3anc 1201 | . . . 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 947 wceq 1316 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 caddc 7591 cmul 7593 clt 7768 cle 7769 cmin 7901 # cap 8311 cdiv 8400 c2 8739 c4 8741 c6 8743 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 |
This theorem is referenced by: fldiv4p1lem1div2 10046 |
Copyright terms: Public domain | W3C validator |