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Theorem 2sqpwodd 11236
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 11234 . . . . . . . 8  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5250 . . . . . . . 8  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5237 . . . . . . . 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 5417 . . . . . 6  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 5919 . . . . . 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 8836 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 8547 . . . . . 6  |-  2  e.  NN
1211a1i 9 . . . . 5  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 8837 . . . 4  |-  ( A  e.  NN  ->  2  e.  ZZ )
1410, 13zmulcld 8844 . . 3  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ )
15 dvdsmul2 10901 . . . 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 10958 . . . . 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 589 . . 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 5918 . . . . . . . . . . 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 3841 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2423notbid 627 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2524, 1elrab2 2772 . . . . . . . . . . 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 10072 . . . . . . . 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 11191 . . . . . . . . . . 11  |-  2  e.  Prime
3227nnzd 8837 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
33 euclemma 11207 . . . . . . . . . . . 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 709 . . . . . . . . . . . 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 1278 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3730, 36mtbird 633 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3827nncnd 8408 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3938sqvald 10048 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
4039breq2d 3849 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
4137, 40mtbird 633 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
42 breq2 3841 . . . . . . . . . 10  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4342notbid 627 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4443, 1elrab2 2772 . . . . . . . 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 8696 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  NN0 )
479, 46nn0mulcld 8701 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
48 peano2nn0 8683 . . . . . . . 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 4457 . . . . . . 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 8408 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  2  e.  CC )
5352, 47expp1d 10052 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 ) )
5452, 47expcld 10051 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  CC )
5554, 52mulcomd 7488 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) ) ) )
5652, 46, 9expmuld 10054 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
5756oveq2d 5650 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2124 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5958oveq1d 5649 . . . . . . . 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 10073 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  e.  NN )
6160, 28nnmulcld 8442 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 2 ^ (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  e.  NN )
62 oveq2 5642 . . . . . . . . . 10  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
63 oveq2 5642 . . . . . . . . . . 11  |-  ( y  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  -> 
( 2 ^ y
)  =  ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
6463oveq1d 5649 . . . . . . . . . 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 5761 . . . . . . . . 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 1174 . . . . . . . 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 5539 . . . . . . . . . . . . . . . 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 5927 . . . . . . . . . . . . . . . . 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 5293 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
7268, 71eqtr3d 2122 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
73 df-ov 5637 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
7472, 73syl6eqr 2138 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
7512, 9nnexpcld 10073 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
7675, 27nnmulcld 8442 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
77 oveq2 5642 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
78 oveq2 5642 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
7978oveq1d 5649 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8077, 79, 2ovmpt2g 5761 . . . . . . . . . . . . . 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 1174 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8274, 81eqtrd 2120 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8382oveq1d 5649 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
8475nncnd 8408 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
8584, 38sqmuld 10063 . . . . . . . . . . 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 2120 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
8786oveq2d 5650 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( 2  x.  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2165 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  e.  CC )
8928nncnd 8408 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  CC )
9052, 88, 89mulassd 7490 . . . . . . . . 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 2123 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2131 . . . . . . 7  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
93 df-ov 5637 . . . . . . 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 2137 . . . . . 6  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) ) )
95 f1ocnvfv 5540 . . . . . . 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 5293 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( 2nd `  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
99 op2ndg 5904 . . . . 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 2120 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
102101breq2d 3849 . 2  |-  ( A  e.  NN  ->  (
2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
10320, 102mtbird 633 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 664    /\ w3a 924    = wceq 1289    e. wcel 1438   {crab 2363   <.cop 3444   class class class wbr 3837    X. cxp 4426   `'ccnv 4427   -->wf 4998   -1-1-onto->wf1o 5001   ` cfv 5002  (class class class)co 5634    |-> cmpt2 5636   1stc1st 5891   2ndc2nd 5892   CCcc 7327   1c1 7330    + caddc 7332    x. cmul 7334   NNcn 8394   2c2 8444   NN0cn0 8643   ZZcz 8720   ^cexp 9919    || cdvds 10878   Primecprime 11171
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 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-xor 1312  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 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-1o 6163  df-2o 6164  df-er 6272  df-en 6438  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-q 9074  df-rp 9104  df-fz 9394  df-fzo 9519  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879  df-gcd 11021  df-prm 11172
This theorem is referenced by:  sqne2sq  11237
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