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Theorem 2sqpwodd 10779
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 10777 . . . . . . . 8  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5191 . . . . . . . 8  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5178 . . . . . . . 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 5354 . . . . . 6  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 5845 . . . . . 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 8618 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 8330 . . . . . 6  |-  2  e.  NN
1211a1i 9 . . . . 5  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 8619 . . . 4  |-  ( A  e.  NN  ->  2  e.  ZZ )
1410, 13zmulcld 8626 . . 3  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ )
15 dvdsmul2 10444 . . . 4  |-  ( ( ( 2nd `  ( `' F `  A ) )  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
1610, 13, 15syl2anc 403 . . 3  |-  ( A  e.  NN  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
17 oddp1even 10501 . . . . 5  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  ( -.  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
1817biimprd 156 . . . 4  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  (
2  ||  ( (
( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  ->  -.  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
1918con2d 587 . . 3  |-  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ  ->  (
2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  ->  -.  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
2014, 16, 19sylc 61 . 2  |-  ( A  e.  NN  ->  -.  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
21 xp1st 5844 . . . . . . . . . . 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 3809 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2423notbid 625 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2524, 1elrab2 2760 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  <->  ( ( 1st `  ( `' F `  A ) )  e.  NN  /\  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2625simplbi 268 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  -> 
( 1st `  ( `' F `  A ) )  e.  NN )
2722, 26syl 14 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  NN )
2827nnsqcld 9793 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN )
2925simprbi 269 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
3022, 29syl 14 . . . . . . . . . 10  |-  ( A  e.  NN  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
31 2prm 10734 . . . . . . . . . . 11  |-  2  e.  Prime
3227nnzd 8619 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
33 euclemma 10750 . . . . . . . . . . . 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 707 . . . . . . . . . . . 12  |-  ( ( 2  ||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) )
3533, 34syl6bb 194 . . . . . . . . . . 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 1274 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3730, 36mtbird 631 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3827nncnd 8190 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3938sqvald 9769 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
4039breq2d 3817 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
4137, 40mtbird 631 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
42 breq2 3809 . . . . . . . . . 10  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4342notbid 625 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4443, 1elrab2 2760 . . . . . . . 8  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  <->  ( (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN  /\  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
4528, 41, 44sylanbrc 408 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J )
4612nnnn0d 8478 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  NN0 )
479, 46nn0mulcld 8483 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
48 peano2nn0 8465 . . . . . . . 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 4420 . . . . . . 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 408 . . . . . 6  |-  ( A  e.  NN  ->  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 ) )
5212nncnd 8190 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  2  e.  CC )
5352, 47expp1d 9773 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 ) )
5452, 47expcld 9772 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  CC )
5554, 52mulcomd 7272 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) ) ) )
5652, 46, 9expmuld 9775 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
5756oveq2d 5580 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2119 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5958oveq1d 5579 . . . . . . . 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 9794 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  e.  NN )
6160, 28nnmulcld 8224 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 2 ^ (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  e.  NN )
62 oveq2 5572 . . . . . . . . . 10  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
63 oveq2 5572 . . . . . . . . . . 11  |-  ( y  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  -> 
( 2 ^ y
)  =  ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
6463oveq1d 5579 . . . . . . . . . 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, 2ovmpt2g 5687 . . . . . . . . 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 1170 . . . . . . . 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 5470 . . . . . . . . . . . . . . . 16  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  A  e.  NN )  ->  ( F `  ( `' F `  A ) )  =  A )
683, 67mpan 415 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  A )
69 1st2nd2 5853 . . . . . . . . . . . . . . . . 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 5234 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
7268, 71eqtr3d 2117 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
73 df-ov 5567 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
7472, 73syl6eqr 2133 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
7512, 9nnexpcld 9794 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
7675, 27nnmulcld 8224 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
77 oveq2 5572 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
78 oveq2 5572 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
7978oveq1d 5579 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8077, 79, 2ovmpt2g 5687 . . . . . . . . . . . . . 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 1170 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8274, 81eqtrd 2115 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8382oveq1d 5579 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
8475nncnd 8190 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
8584, 38sqmuld 9784 . . . . . . . . . . 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 2115 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
8786oveq2d 5580 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( 2  x.  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2160 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  e.  CC )
8928nncnd 8190 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  CC )
9052, 88, 89mulassd 7274 . . . . . . . . 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 2118 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2126 . . . . . . 7  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
93 df-ov 5567 . . . . . . 7  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)
9492, 93syl6req 2132 . . . . . 6  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) ) )
95 f1ocnvfv 5471 . . . . . . 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 415 . . . . . 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 61 . . . . 5  |-  ( A  e.  NN  ->  ( `' F `  ( 2  x.  ( A ^
2 ) ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)
9897fveq2d 5234 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( 2nd `  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
99 op2ndg 5830 . . . . 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 403 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
10198, 100eqtrd 2115 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
102101breq2d 3817 . 2  |-  ( A  e.  NN  ->  (
2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
10320, 102mtbird 631 1  |-  ( A  e.  NN  ->  -.  2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^
2 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   {crab 2357   <.cop 3419   class class class wbr 3805    X. cxp 4389   `'ccnv 4390   -->wf 4948   -1-1-onto->wf1o 4951   ` cfv 4952  (class class class)co 5564    |-> cmpt2 5566   1stc1st 5817   2ndc2nd 5818   CCcc 7111   1c1 7114    + caddc 7116    x. cmul 7118   NNcn 8176   2c2 8226   NN0cn0 8425   ZZcz 8502   ^cexp 9642    || cdvds 10421   Primecprime 10714
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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227  ax-caucvg 7228
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-1o 6086  df-2o 6087  df-er 6194  df-en 6310  df-sup 6492  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-3 8236  df-4 8237  df-n0 8426  df-z 8503  df-uz 8771  df-q 8856  df-rp 8886  df-fz 9176  df-fzo 9300  df-fl 9422  df-mod 9475  df-iseq 9592  df-iexp 9643  df-cj 9948  df-re 9949  df-im 9950  df-rsqrt 10103  df-abs 10104  df-dvds 10422  df-gcd 10564  df-prm 10715
This theorem is referenced by:  sqne2sq  10780
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