ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sqoddm1div8 Unicode version

Theorem sqoddm1div8 10670
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.)
Assertion
Ref Expression
sqoddm1div8  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( M ^ 2 )  -  1 )  / 
8 )  =  ( ( N  x.  ( N  +  1 ) )  /  2 ) )

Proof of Theorem sqoddm1div8
StepHypRef Expression
1 oveq1 5881 . . . . . 6  |-  ( M  =  ( ( 2  x.  N )  +  1 )  ->  ( M ^ 2 )  =  ( ( ( 2  x.  N )  +  1 ) ^ 2 ) )
2 2z 9279 . . . . . . . . . 10  |-  2  e.  ZZ
32a1i 9 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  2  e.  ZZ )
4 id 19 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  N  e.  ZZ )
53, 4zmulcld 9379 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  ZZ )
65zcnd 9374 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  CC )
7 binom21 10629 . . . . . . 7  |-  ( ( 2  x.  N )  e.  CC  ->  (
( ( 2  x.  N )  +  1 ) ^ 2 )  =  ( ( ( ( 2  x.  N
) ^ 2 )  +  ( 2  x.  ( 2  x.  N
) ) )  +  1 ) )
86, 7syl 14 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  N )  +  1 ) ^ 2 )  =  ( ( ( ( 2  x.  N
) ^ 2 )  +  ( 2  x.  ( 2  x.  N
) ) )  +  1 ) )
91, 8sylan9eqr 2232 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( M ^
2 )  =  ( ( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 ) )
109oveq1d 5889 . . . 4  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( M ^ 2 )  - 
1 )  =  ( ( ( ( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  (
2  x.  N ) ) )  +  1 )  -  1 ) )
11 2cnd 8990 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  2  e.  CC )
12 zcn 9256 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
1311, 12sqmuld 10662 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) ) )
14 sq2 10612 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
1514a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2 ^ 2 )  =  4 )
1615oveq1d 5889 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2 ^ 2 )  x.  ( N ^ 2 ) )  =  ( 4  x.  ( N ^ 2 ) ) )
1713, 16eqtrd 2210 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( 4  x.  ( N ^ 2 ) ) )
18 mulass 7941 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
( 2  x.  2 )  x.  N )  =  ( 2  x.  ( 2  x.  N
) ) )
1918eqcomd 2183 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
2011, 11, 12, 19syl3anc 1238 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
21 2t2e4 9071 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
2221a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2  x.  2 )  =  4 )
2322oveq1d 5889 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  2 )  x.  N )  =  ( 4  x.  N ) )
2420, 23eqtrd 2210 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( 4  x.  N ) )
2517, 24oveq12d 5892 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  ( 2  x.  N ) ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
2625oveq1d 5889 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  +  1 ) )
2726oveq1d 5889 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ( ( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  (
2  x.  N ) ) )  +  1 )  -  1 )  =  ( ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  +  1 )  -  1 ) )
28 4z 9281 . . . . . . . . . . 11  |-  4  e.  ZZ
2928a1i 9 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  4  e.  ZZ )
30 zsqcl 10587 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
3129, 30zmulcld 9379 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  ZZ )
3231zcnd 9374 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  CC )
3329, 4zmulcld 9379 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  ZZ )
3433zcnd 9374 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  CC )
3532, 34addcld 7975 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  e.  CC )
36 pncan1 8332 . . . . . . 7  |-  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  e.  CC  ->  ( (
( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  +  1 )  -  1 )  =  ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) ) )
3735, 36syl 14 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) )  +  1 )  -  1 )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
3827, 37eqtrd 2210 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( ( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  (
2  x.  N ) ) )  +  1 )  -  1 )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
3938adantr 276 . . . 4  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( ( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 )  - 
1 )  =  ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) ) )
4010, 39eqtrd 2210 . . 3  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( M ^ 2 )  - 
1 )  =  ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) ) )
4140oveq1d 5889 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( M ^ 2 )  -  1 )  / 
8 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  /  8 ) )
42 4cn 8995 . . . . . . 7  |-  4  e.  CC
4342a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  4  e.  CC )
4430zcnd 9374 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  CC )
4543, 44, 12adddid 7980 . . . . 5  |-  ( N  e.  ZZ  ->  (
4  x.  ( ( N ^ 2 )  +  N ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
4645eqcomd 2183 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  =  ( 4  x.  (
( N ^ 2 )  +  N ) ) )
4746oveq1d 5889 . . 3  |-  ( N  e.  ZZ  ->  (
( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  /  8 )  =  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
8 ) )
4847adantr 276 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  / 
8 )  =  ( ( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 ) )
49 4t2e8 9075 . . . . . . 7  |-  ( 4  x.  2 )  =  8
5049a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  (
4  x.  2 )  =  8 )
5150eqcomd 2183 . . . . 5  |-  ( N  e.  ZZ  ->  8  =  ( 4  x.  2 ) )
5251oveq2d 5890 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 )  =  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
( 4  x.  2 ) ) )
5330, 4zaddcld 9377 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  ZZ )
5453zcnd 9374 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  CC )
55 2ap0 9010 . . . . . 6  |-  2 #  0
5655a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  2 #  0 )
57 4ap0 9016 . . . . . 6  |-  4 #  0
5857a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  4 #  0 )
5954, 11, 43, 56, 58divcanap5d 8772 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  ( 4  x.  2 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
6012sqvald 10647 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  =  ( N  x.  N
) )
6160oveq1d 5889 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( ( N  x.  N )  +  N ) )
6212mulridd 7973 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
6362eqcomd 2183 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  =  ( N  x.  1 ) )
6463oveq2d 5890 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  N )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
65 1cnd 7972 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
66 adddi 7942 . . . . . . . 8  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
6766eqcomd 2183 . . . . . . 7  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6812, 12, 65, 67syl3anc 1238 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6961, 64, 683eqtrd 2214 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( N  x.  ( N  +  1
) ) )
7069oveq1d 5889 . . . 4  |-  ( N  e.  ZZ  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( ( N  x.  ( N  + 
1 ) )  / 
2 ) )
7152, 59, 703eqtrd 2214 . . 3  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 )  =  ( ( N  x.  ( N  + 
1 ) )  / 
2 ) )
7271adantr 276 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
8 )  =  ( ( N  x.  ( N  +  1 ) )  /  2 ) )
7341, 48, 723eqtrd 2214 1  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( M ^ 2 )  -  1 )  / 
8 )  =  ( ( N  x.  ( N  +  1 ) )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4003  (class class class)co 5874   CCcc 7808   0cc0 7810   1c1 7811    + caddc 7813    x. cmul 7815    - cmin 8126   # cap 8536    / cdiv 8627   2c2 8968   4c4 8970   8c8 8974   ZZcz 9251   ^cexp 10516
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-5 8979  df-6 8980  df-7 8981  df-8 8982  df-n0 9175  df-z 9252  df-uz 9527  df-seqfrec 10443  df-exp 10517
This theorem is referenced by:  sqoddm1div8z  11885
  Copyright terms: Public domain W3C validator