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Theorem sqoddm1div8 10106
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 5659 . . . . . 6  |-  ( M  =  ( ( 2  x.  N )  +  1 )  ->  ( M ^ 2 )  =  ( ( ( 2  x.  N )  +  1 ) ^ 2 ) )
2 2z 8778 . . . . . . . . . 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 8874 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  ZZ )
65zcnd 8869 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  CC )
7 binom21 10066 . . . . . . 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 2142 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( M ^
2 )  =  ( ( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 ) )
109oveq1d 5667 . . . 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 8495 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  2  e.  CC )
12 zcn 8755 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
1311, 12sqmuld 10098 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) ) )
14 sq2 10050 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
1514a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2 ^ 2 )  =  4 )
1615oveq1d 5667 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2 ^ 2 )  x.  ( N ^ 2 ) )  =  ( 4  x.  ( N ^ 2 ) ) )
1713, 16eqtrd 2120 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( 4  x.  ( N ^ 2 ) ) )
18 mulass 7473 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
( 2  x.  2 )  x.  N )  =  ( 2  x.  ( 2  x.  N
) ) )
1918eqcomd 2093 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
2011, 11, 12, 19syl3anc 1174 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
21 2t2e4 8570 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
2221a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2  x.  2 )  =  4 )
2322oveq1d 5667 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  2 )  x.  N )  =  ( 4  x.  N ) )
2420, 23eqtrd 2120 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( 4  x.  N ) )
2517, 24oveq12d 5670 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  ( 2  x.  N ) ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
2625oveq1d 5667 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  +  1 ) )
2726oveq1d 5667 . . . . . 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 8780 . . . . . . . . . . 11  |-  4  e.  ZZ
2928a1i 9 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  4  e.  ZZ )
30 zsqcl 10025 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
3129, 30zmulcld 8874 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  ZZ )
3231zcnd 8869 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  CC )
3329, 4zmulcld 8874 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  ZZ )
3433zcnd 8869 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  CC )
3532, 34addcld 7507 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  e.  CC )
36 pncan1 7855 . . . . . . 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 2120 . . . . 5  |-  ( N  e.  ZZ  ->  (
( ( ( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  (
2  x.  N ) ) )  +  1 )  -  1 )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
3938adantr 270 . . . 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 2120 . . 3  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( M ^ 2 )  - 
1 )  =  ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) ) )
4140oveq1d 5667 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( M ^ 2 )  -  1 )  / 
8 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  /  8 ) )
42 4cn 8500 . . . . . . 7  |-  4  e.  CC
4342a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  4  e.  CC )
4430zcnd 8869 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  CC )
4543, 44, 12adddid 7512 . . . . 5  |-  ( N  e.  ZZ  ->  (
4  x.  ( ( N ^ 2 )  +  N ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
4645eqcomd 2093 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  =  ( 4  x.  (
( N ^ 2 )  +  N ) ) )
4746oveq1d 5667 . . 3  |-  ( N  e.  ZZ  ->  (
( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  /  8 )  =  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
8 ) )
4847adantr 270 . 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 8574 . . . . . . 7  |-  ( 4  x.  2 )  =  8
5049a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  (
4  x.  2 )  =  8 )
5150eqcomd 2093 . . . . 5  |-  ( N  e.  ZZ  ->  8  =  ( 4  x.  2 ) )
5251oveq2d 5668 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 )  =  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
( 4  x.  2 ) ) )
5330, 4zaddcld 8872 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  ZZ )
5453zcnd 8869 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  CC )
55 2ap0 8515 . . . . . 6  |-  2 #  0
5655a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  2 #  0 )
57 4ap0 8521 . . . . . 6  |-  4 #  0
5857a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  4 #  0 )
5954, 11, 43, 56, 58divcanap5d 8284 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  ( 4  x.  2 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
6012sqvald 10083 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  =  ( N  x.  N
) )
6160oveq1d 5667 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( ( N  x.  N )  +  N ) )
6212mulid1d 7505 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
6362eqcomd 2093 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  =  ( N  x.  1 ) )
6463oveq2d 5668 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  N )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
65 1cnd 7504 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
66 adddi 7474 . . . . . . . 8  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
6766eqcomd 2093 . . . . . . 7  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6812, 12, 65, 67syl3anc 1174 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6961, 64, 683eqtrd 2124 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( N  x.  ( N  +  1
) ) )
7069oveq1d 5667 . . . 4  |-  ( N  e.  ZZ  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( ( N  x.  ( N  + 
1 ) )  / 
2 ) )
7152, 59, 703eqtrd 2124 . . 3  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 )  =  ( ( N  x.  ( N  + 
1 ) )  / 
2 ) )
7271adantr 270 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
8 )  =  ( ( N  x.  ( N  +  1 ) )  /  2 ) )
7341, 48, 723eqtrd 2124 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 102    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   CCcc 7348   0cc0 7350   1c1 7351    + caddc 7353    x. cmul 7355    - cmin 7653   # cap 8058    / cdiv 8139   2c2 8473   4c4 8475   8c8 8479   ZZcz 8750   ^cexp 9954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-mulrcl 7444  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-precex 7455  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461  ax-pre-mulgt0 7462  ax-pre-mulext 7463
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-reap 8052  df-ap 8059  df-div 8140  df-inn 8423  df-2 8481  df-3 8482  df-4 8483  df-5 8484  df-6 8485  df-7 8486  df-8 8487  df-n0 8674  df-z 8751  df-uz 9020  df-iseq 9853  df-seq3 9854  df-exp 9955
This theorem is referenced by:  sqoddm1div8z  11164
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