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Theorem sqoddm1div8 9711
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 5544 . . . . . 6  |-  ( M  =  ( ( 2  x.  N )  +  1 )  ->  ( M ^ 2 )  =  ( ( ( 2  x.  N )  +  1 ) ^ 2 ) )
2 2z 8449 . . . . . . . . . 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 8545 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  ZZ )
65zcnd 8540 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
2  x.  N )  e.  CC )
7 binom21 9672 . . . . . . 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 2136 . . . . 5  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( M ^
2 )  =  ( ( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 ) )
109oveq1d 5552 . . . 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 8168 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  2  e.  CC )
12 zcn 8426 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  CC )
1311, 12sqmuld 9703 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( N ^ 2 ) ) )
14 sq2 9657 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
1514a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2 ^ 2 )  =  4 )
1615oveq1d 5552 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2 ^ 2 )  x.  ( N ^ 2 ) )  =  ( 4  x.  ( N ^ 2 ) ) )
1713, 16eqtrd 2114 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 2  x.  N
) ^ 2 )  =  ( 4  x.  ( N ^ 2 ) ) )
18 mulass 7155 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
( 2  x.  2 )  x.  N )  =  ( 2  x.  ( 2  x.  N
) ) )
1918eqcomd 2087 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  2  e.  CC  /\  N  e.  CC )  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
2011, 11, 12, 19syl3anc 1170 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( ( 2  x.  2 )  x.  N ) )
21 2t2e4 8242 . . . . . . . . . . . 12  |-  ( 2  x.  2 )  =  4
2221a1i 9 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
2  x.  2 )  =  4 )
2322oveq1d 5552 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (
( 2  x.  2 )  x.  N )  =  ( 4  x.  N ) )
2420, 23eqtrd 2114 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
2  x.  ( 2  x.  N ) )  =  ( 4  x.  N ) )
2517, 24oveq12d 5555 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
( ( 2  x.  N ) ^ 2 )  +  ( 2  x.  ( 2  x.  N ) ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
2625oveq1d 5552 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( ( ( 2  x.  N ) ^
2 )  +  ( 2  x.  ( 2  x.  N ) ) )  +  1 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  +  1 ) )
2726oveq1d 5552 . . . . . 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 8451 . . . . . . . . . . 11  |-  4  e.  ZZ
2928a1i 9 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  4  e.  ZZ )
30 zsqcl 9632 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  ZZ )
3129, 30zmulcld 8545 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  ZZ )
3231zcnd 8540 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  ( N ^ 2 ) )  e.  CC )
3329, 4zmulcld 8545 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  ZZ )
3433zcnd 8540 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
4  x.  N )  e.  CC )
3532, 34addcld 7189 . . . . . . 7  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  e.  CC )
36 pncan1 7537 . . . . . . 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 2114 . . . . 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 2114 . . 3  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( M ^ 2 )  - 
1 )  =  ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) ) )
4140oveq1d 5552 . 2  |-  ( ( N  e.  ZZ  /\  M  =  ( (
2  x.  N )  +  1 ) )  ->  ( ( ( M ^ 2 )  -  1 )  / 
8 )  =  ( ( ( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  /  8 ) )
42 4cn 8173 . . . . . . 7  |-  4  e.  CC
4342a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  4  e.  CC )
4430zcnd 8540 . . . . . 6  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  e.  CC )
4543, 44, 12adddid 7194 . . . . 5  |-  ( N  e.  ZZ  ->  (
4  x.  ( ( N ^ 2 )  +  N ) )  =  ( ( 4  x.  ( N ^
2 ) )  +  ( 4  x.  N
) ) )
4645eqcomd 2087 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  ( N ^ 2 ) )  +  ( 4  x.  N ) )  =  ( 4  x.  (
( N ^ 2 )  +  N ) ) )
4746oveq1d 5552 . . 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 8246 . . . . . . 7  |-  ( 4  x.  2 )  =  8
5049a1i 9 . . . . . 6  |-  ( N  e.  ZZ  ->  (
4  x.  2 )  =  8 )
5150eqcomd 2087 . . . . 5  |-  ( N  e.  ZZ  ->  8  =  ( 4  x.  2 ) )
5251oveq2d 5553 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  8 )  =  ( ( 4  x.  ( ( N ^ 2 )  +  N ) )  / 
( 4  x.  2 ) ) )
5330, 4zaddcld 8543 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  ZZ )
5453zcnd 8540 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  e.  CC )
55 2ap0 8188 . . . . . 6  |-  2 #  0
5655a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  2 #  0 )
57 4ap0 8194 . . . . . 6  |-  4 #  0
5857a1i 9 . . . . 5  |-  ( N  e.  ZZ  ->  4 #  0 )
5954, 11, 43, 56, 58divcanap5d 7959 . . . 4  |-  ( N  e.  ZZ  ->  (
( 4  x.  (
( N ^ 2 )  +  N ) )  /  ( 4  x.  2 ) )  =  ( ( ( N ^ 2 )  +  N )  / 
2 ) )
6012sqvald 9688 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( N ^ 2 )  =  ( N  x.  N
) )
6160oveq1d 5552 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( ( N  x.  N )  +  N ) )
6212mulid1d 7187 . . . . . . . 8  |-  ( N  e.  ZZ  ->  ( N  x.  1 )  =  N )
6362eqcomd 2087 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  =  ( N  x.  1 ) )
6463oveq2d 5553 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  N )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
65 1cnd 7186 . . . . . . 7  |-  ( N  e.  ZZ  ->  1  e.  CC )
66 adddi 7156 . . . . . . . 8  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( N  x.  ( N  +  1 ) )  =  ( ( N  x.  N )  +  ( N  x.  1 ) ) )
6766eqcomd 2087 . . . . . . 7  |-  ( ( N  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6812, 12, 65, 67syl3anc 1170 . . . . . 6  |-  ( N  e.  ZZ  ->  (
( N  x.  N
)  +  ( N  x.  1 ) )  =  ( N  x.  ( N  +  1
) ) )
6961, 64, 683eqtrd 2118 . . . . 5  |-  ( N  e.  ZZ  ->  (
( N ^ 2 )  +  N )  =  ( N  x.  ( N  +  1
) ) )
7069oveq1d 5552 . . . 4  |-  ( N  e.  ZZ  ->  (
( ( N ^
2 )  +  N
)  /  2 )  =  ( ( N  x.  ( N  + 
1 ) )  / 
2 ) )
7152, 59, 703eqtrd 2118 . . 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 2118 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 920    = wceq 1285    e. wcel 1434   class class class wbr 3787  (class class class)co 5537   CCcc 7030   0cc0 7032   1c1 7033    + caddc 7035    x. cmul 7037    - cmin 7335   # cap 7737    / cdiv 7816   2c2 8145   4c4 8147   8c8 8151   ZZcz 8421   ^cexp 9561
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-if 3354  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-2 8154  df-3 8155  df-4 8156  df-5 8157  df-6 8158  df-7 8159  df-8 8160  df-n0 8345  df-z 8422  df-uz 8690  df-iseq 9511  df-iexp 9562
This theorem is referenced by:  sqoddm1div8z  10419
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