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Theorem 2sqpwodd 12130
Description: The greatest power of two dividing twice the square of an integer is an odd power of two. (Contributed by Jim Kingdon, 17-Nov-2021.)
Hypotheses
Ref Expression
oddpwdc.j  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
oddpwdc.f  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
Assertion
Ref Expression
2sqpwodd  |-  ( A  e.  NN  ->  -.  2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^
2 ) ) ) ) )
Distinct variable groups:    x, y, z   
x, J, y    x, A, y, z    x, F, y, z
Allowed substitution hint:    J( z)

Proof of Theorem 2sqpwodd
StepHypRef Expression
1 oddpwdc.j . . . . . . . . 9  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
2 oddpwdc.f . . . . . . . . 9  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
31, 2oddpwdc 12128 . . . . . . . 8  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5455 . . . . . . . 8  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5442 . . . . . . . 8  |-  ( `' F : NN -1-1-onto-> ( J  X.  NN0 )  ->  `' F : NN
--> ( J  X.  NN0 ) )
63, 4, 5mp2b 8 . . . . . . 7  |-  `' F : NN --> ( J  X.  NN0 )
76ffvelrni 5630 . . . . . 6  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 6145 . . . . . 6  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( 2nd `  ( `' F `  A ) )  e. 
NN0 )
97, 8syl 14 . . . . 5  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e. 
NN0 )
109nn0zd 9332 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 9039 . . . . . 6  |-  2  e.  NN
1211a1i 9 . . . . 5  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 9333 . . . 4  |-  ( A  e.  NN  ->  2  e.  ZZ )
1410, 13zmulcld 9340 . . 3  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ )
15 dvdsmul2 11776 . . . 4  |-  ( ( ( 2nd `  ( `' F `  A ) )  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
1610, 13, 15syl2anc 409 . . 3  |-  ( A  e.  NN  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
17 oddp1even 11835 . . . . 5  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  ( -.  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
1817biimprd 157 . . . 4  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  (
2  ||  ( (
( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  ->  -.  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
1918con2d 619 . . 3  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  (
2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  ->  -.  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
2014, 16, 19sylc 62 . 2  |-  ( A  e.  NN  ->  -.  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
21 xp1st 6144 . . . . . . . . . . 11  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( 1st `  ( `' F `  A ) )  e.  J )
227, 21syl 14 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  J )
23 breq2 3993 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2423notbid 662 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2524, 1elrab2 2889 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  <->  ( ( 1st `  ( `' F `  A ) )  e.  NN  /\  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2625simplbi 272 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  -> 
( 1st `  ( `' F `  A ) )  e.  NN )
2722, 26syl 14 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  NN )
2827nnsqcld 10630 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN )
2925simprbi 273 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
3022, 29syl 14 . . . . . . . . . 10  |-  ( A  e.  NN  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
31 2prm 12081 . . . . . . . . . . 11  |-  2  e.  Prime
3227nnzd 9333 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
33 euclemma 12100 . . . . . . . . . . . 12  |-  ( ( 2  e.  Prime  /\  ( 1st `  ( `' F `  A ) )  e.  ZZ  /\  ( 1st `  ( `' F `  A ) )  e.  ZZ )  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  ( 2 
||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) ) ) )
34 oridm 752 . . . . . . . . . . . 12  |-  ( ( 2  ||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) )
3533, 34bitrdi 195 . . . . . . . . . . 11  |-  ( ( 2  e.  Prime  /\  ( 1st `  ( `' F `  A ) )  e.  ZZ  /\  ( 1st `  ( `' F `  A ) )  e.  ZZ )  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3631, 32, 32, 35mp3an2i 1337 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3730, 36mtbird 668 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3827nncnd 8892 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3938sqvald 10606 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
4039breq2d 4001 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
4137, 40mtbird 668 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
42 breq2 3993 . . . . . . . . . 10  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4342notbid 662 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4443, 1elrab2 2889 . . . . . . . 8  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  <->  ( (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN  /\  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
4528, 41, 44sylanbrc 415 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J )
4612nnnn0d 9188 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  NN0 )
479, 46nn0mulcld 9193 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
48 peano2nn0 9175 . . . . . . . 8  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0  ->  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  e. 
NN0 )
4947, 48syl 14 . . . . . . 7  |-  ( A  e.  NN  ->  (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  e. 
NN0 )
50 opelxp 4641 . . . . . . 7  |-  ( <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 ) 
<->  ( ( ( 1st `  ( `' F `  A ) ) ^
2 )  e.  J  /\  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  e.  NN0 )
)
5145, 49, 50sylanbrc 415 . . . . . 6  |-  ( A  e.  NN  ->  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 ) )
5212nncnd 8892 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  2  e.  CC )
5352, 47expp1d 10610 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 ) )
5452, 47expcld 10609 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  CC )
5554, 52mulcomd 7941 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) ) ) )
5652, 46, 9expmuld 10612 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
5756oveq2d 5869 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2207 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5958oveq1d 5868 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2 ^ (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  =  ( ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
6012, 49nnexpcld 10631 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  e.  NN )
6160, 28nnmulcld 8927 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 2 ^ (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  e.  NN )
62 oveq2 5861 . . . . . . . . . 10  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
63 oveq2 5861 . . . . . . . . . . 11  |-  ( y  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  -> 
( 2 ^ y
)  =  ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
6463oveq1d 5868 . . . . . . . . . 10  |-  ( y  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  -> 
( ( 2 ^ y )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) )  =  ( ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
6562, 64, 2ovmpog 5987 . . . . . . . . 9  |-  ( ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  /\  (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  e. 
NN0  /\  ( (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  e.  NN )  ->  (
( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
6645, 49, 61, 65syl3anc 1233 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
67 f1ocnvfv2 5757 . . . . . . . . . . . . . . . 16  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  A  e.  NN )  ->  ( F `  ( `' F `  A ) )  =  A )
683, 67mpan 422 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  A )
69 1st2nd2 6154 . . . . . . . . . . . . . . . . 17  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( `' F `  A )  =  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
707, 69syl 14 . . . . . . . . . . . . . . . 16  |-  ( A  e.  NN  ->  ( `' F `  A )  =  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
7170fveq2d 5500 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
7268, 71eqtr3d 2205 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
73 df-ov 5856 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
7472, 73eqtr4di 2221 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
7512, 9nnexpcld 10631 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
7675, 27nnmulcld 8927 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
77 oveq2 5861 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
78 oveq2 5861 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
7978oveq1d 5868 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8077, 79, 2ovmpog 5987 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( `' F `  A ) )  e.  J  /\  ( 2nd `  ( `' F `  A ) )  e.  NN0  /\  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )  -> 
( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8122, 9, 76, 80syl3anc 1233 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8274, 81eqtrd 2203 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8382oveq1d 5868 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
8475nncnd 8892 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
8584, 38sqmuld 10621 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
8683, 85eqtrd 2203 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
8786oveq2d 5869 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( 2  x.  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2248 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  e.  CC )
8928nncnd 8892 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  CC )
9052, 88, 89mulassd 7943 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 2  x.  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  =  ( 2  x.  (
( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) ) )
9187, 90eqtr4d 2206 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2214 . . . . . . 7  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
93 df-ov 5856 . . . . . . 7  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)
9492, 93eqtr2di 2220 . . . . . 6  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) ) )
95 f1ocnvfv 5758 . . . . . . 7  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 ) )  ->  (
( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) )  -> 
( `' F `  ( 2  x.  ( A ^ 2 ) ) )  =  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
963, 95mpan 422 . . . . . 6  |-  ( <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 )  ->  ( ( F `
 <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) )  -> 
( `' F `  ( 2  x.  ( A ^ 2 ) ) )  =  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
9751, 94, 96sylc 62 . . . . 5  |-  ( A  e.  NN  ->  ( `' F `  ( 2  x.  ( A ^
2 ) ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)
9897fveq2d 5500 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( 2nd `  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
99 op2ndg 6130 . . . . 5  |-  ( ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  /\  (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  e. 
NN0 )  ->  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
10045, 49, 99syl2anc 409 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
10198, 100eqtrd 2203 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
102101breq2d 4001 . 2  |-  ( A  e.  NN  ->  (
2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
10320, 102mtbird 668 1  |-  ( A  e.  NN  ->  -.  2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^
2 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 703    /\ w3a 973    = wceq 1348    e. wcel 2141   {crab 2452   <.cop 3586   class class class wbr 3989    X. cxp 4609   `'ccnv 4610   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853    e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   CCcc 7772   1c1 7775    + caddc 7777    x. cmul 7779   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ^cexp 10475    || cdvds 11749   Primecprime 12061
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062
This theorem is referenced by:  sqne2sq  12131
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