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Mirrors > Home > ILE Home > Th. List > sqoddm1div8 | Unicode version |
Description: A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) |
Ref | Expression |
---|---|
sqoddm1div8 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5774 | . . . . . 6 | |
2 | 2z 9075 | . . . . . . . . . 10 | |
3 | 2 | a1i 9 | . . . . . . . . 9 |
4 | id 19 | . . . . . . . . 9 | |
5 | 3, 4 | zmulcld 9172 | . . . . . . . 8 |
6 | 5 | zcnd 9167 | . . . . . . 7 |
7 | binom21 10397 | . . . . . . 7 | |
8 | 6, 7 | syl 14 | . . . . . 6 |
9 | 1, 8 | sylan9eqr 2192 | . . . . 5 |
10 | 9 | oveq1d 5782 | . . . 4 |
11 | 2cnd 8786 | . . . . . . . . . . 11 | |
12 | zcn 9052 | . . . . . . . . . . 11 | |
13 | 11, 12 | sqmuld 10429 | . . . . . . . . . 10 |
14 | sq2 10381 | . . . . . . . . . . . 12 | |
15 | 14 | a1i 9 | . . . . . . . . . . 11 |
16 | 15 | oveq1d 5782 | . . . . . . . . . 10 |
17 | 13, 16 | eqtrd 2170 | . . . . . . . . 9 |
18 | mulass 7744 | . . . . . . . . . . . 12 | |
19 | 18 | eqcomd 2143 | . . . . . . . . . . 11 |
20 | 11, 11, 12, 19 | syl3anc 1216 | . . . . . . . . . 10 |
21 | 2t2e4 8867 | . . . . . . . . . . . 12 | |
22 | 21 | a1i 9 | . . . . . . . . . . 11 |
23 | 22 | oveq1d 5782 | . . . . . . . . . 10 |
24 | 20, 23 | eqtrd 2170 | . . . . . . . . 9 |
25 | 17, 24 | oveq12d 5785 | . . . . . . . 8 |
26 | 25 | oveq1d 5782 | . . . . . . 7 |
27 | 26 | oveq1d 5782 | . . . . . 6 |
28 | 4z 9077 | . . . . . . . . . . 11 | |
29 | 28 | a1i 9 | . . . . . . . . . 10 |
30 | zsqcl 10356 | . . . . . . . . . 10 | |
31 | 29, 30 | zmulcld 9172 | . . . . . . . . 9 |
32 | 31 | zcnd 9167 | . . . . . . . 8 |
33 | 29, 4 | zmulcld 9172 | . . . . . . . . 9 |
34 | 33 | zcnd 9167 | . . . . . . . 8 |
35 | 32, 34 | addcld 7778 | . . . . . . 7 |
36 | pncan1 8132 | . . . . . . 7 | |
37 | 35, 36 | syl 14 | . . . . . 6 |
38 | 27, 37 | eqtrd 2170 | . . . . 5 |
39 | 38 | adantr 274 | . . . 4 |
40 | 10, 39 | eqtrd 2170 | . . 3 |
41 | 40 | oveq1d 5782 | . 2 |
42 | 4cn 8791 | . . . . . . 7 | |
43 | 42 | a1i 9 | . . . . . 6 |
44 | 30 | zcnd 9167 | . . . . . 6 |
45 | 43, 44, 12 | adddid 7783 | . . . . 5 |
46 | 45 | eqcomd 2143 | . . . 4 |
47 | 46 | oveq1d 5782 | . . 3 |
48 | 47 | adantr 274 | . 2 |
49 | 4t2e8 8871 | . . . . . . 7 | |
50 | 49 | a1i 9 | . . . . . 6 |
51 | 50 | eqcomd 2143 | . . . . 5 |
52 | 51 | oveq2d 5783 | . . . 4 |
53 | 30, 4 | zaddcld 9170 | . . . . . 6 |
54 | 53 | zcnd 9167 | . . . . 5 |
55 | 2ap0 8806 | . . . . . 6 # | |
56 | 55 | a1i 9 | . . . . 5 # |
57 | 4ap0 8812 | . . . . . 6 # | |
58 | 57 | a1i 9 | . . . . 5 # |
59 | 54, 11, 43, 56, 58 | divcanap5d 8570 | . . . 4 |
60 | 12 | sqvald 10414 | . . . . . . 7 |
61 | 60 | oveq1d 5782 | . . . . . 6 |
62 | 12 | mulid1d 7776 | . . . . . . . 8 |
63 | 62 | eqcomd 2143 | . . . . . . 7 |
64 | 63 | oveq2d 5783 | . . . . . 6 |
65 | 1cnd 7775 | . . . . . . 7 | |
66 | adddi 7745 | . . . . . . . 8 | |
67 | 66 | eqcomd 2143 | . . . . . . 7 |
68 | 12, 12, 65, 67 | syl3anc 1216 | . . . . . 6 |
69 | 61, 64, 68 | 3eqtrd 2174 | . . . . 5 |
70 | 69 | oveq1d 5782 | . . . 4 |
71 | 52, 59, 70 | 3eqtrd 2174 | . . 3 |
72 | 71 | adantr 274 | . 2 |
73 | 41, 48, 72 | 3eqtrd 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 class class class wbr 3924 (class class class)co 5767 cc 7611 cc0 7613 c1 7614 caddc 7616 cmul 7618 cmin 7926 # cap 8336 cdiv 8425 c2 8764 c4 8766 c8 8770 cz 9047 cexp 10285 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-exp 10286 |
This theorem is referenced by: sqoddm1div8z 11572 |
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