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Theorem 2sqpwodd 11888
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 11886 . . . . . . . 8  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5387 . . . . . . . 8  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5374 . . . . . . . 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 5561 . . . . . 6  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 6071 . . . . . 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 9194 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 8904 . . . . . 6  |-  2  e.  NN
1211a1i 9 . . . . 5  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 9195 . . . 4  |-  ( A  e.  NN  ->  2  e.  ZZ )
1410, 13zmulcld 9202 . . 3  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  ZZ )
15 dvdsmul2 11550 . . . 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 11607 . . . . 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 614 . . 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 6070 . . . . . . . . . . 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 3940 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2423notbid 657 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2524, 1elrab2 2846 . . . . . . . . . . 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 10475 . . . . . . . 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 11842 . . . . . . . . . . 11  |-  2  e.  Prime
3227nnzd 9195 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
33 euclemma 11858 . . . . . . . . . . . 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 747 . . . . . . . . . . . 12  |-  ( ( 2  ||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) )
3533, 34syl6bb 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 1321 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3730, 36mtbird 663 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3827nncnd 8757 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3938sqvald 10451 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
4039breq2d 3948 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
4137, 40mtbird 663 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
42 breq2 3940 . . . . . . . . . 10  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4342notbid 657 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4443, 1elrab2 2846 . . . . . . . 8  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  <->  ( (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN  /\  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
4528, 41, 44sylanbrc 414 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J )
4612nnnn0d 9053 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  NN0 )
479, 46nn0mulcld 9058 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
48 peano2nn0 9040 . . . . . . . 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 4576 . . . . . . 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 414 . . . . . 6  |-  ( A  e.  NN  ->  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.  e.  ( J  X.  NN0 ) )
5212nncnd 8757 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  2  e.  CC )
5352, 47expp1d 10455 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 ) )
5452, 47expcld 10454 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  CC )
5554, 52mulcomd 7810 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) ) ) )
5652, 46, 9expmuld 10457 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
5756oveq2d 5797 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2  x.  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2177 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  =  ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) ) )
5958oveq1d 5796 . . . . . . . 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 10476 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  e.  NN )
6160, 28nnmulcld 8792 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 2 ^ (
( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )  e.  NN )
62 oveq2 5789 . . . . . . . . . 10  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
63 oveq2 5789 . . . . . . . . . . 11  |-  ( y  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 )  -> 
( 2 ^ y
)  =  ( 2 ^ ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
6463oveq1d 5796 . . . . . . . . . 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 5912 . . . . . . . . 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 1217 . . . . . . . 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 5686 . . . . . . . . . . . . . . . 16  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  A  e.  NN )  ->  ( F `  ( `' F `  A ) )  =  A )
683, 67mpan 421 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  A )
69 1st2nd2 6080 . . . . . . . . . . . . . . . . 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 5432 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
7268, 71eqtr3d 2175 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
73 df-ov 5784 . . . . . . . . . . . . . 14  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
7472, 73eqtr4di 2191 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
7512, 9nnexpcld 10476 . . . . . . . . . . . . . . 15  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
7675, 27nnmulcld 8792 . . . . . . . . . . . . . 14  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
77 oveq2 5789 . . . . . . . . . . . . . . 15  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
78 oveq2 5789 . . . . . . . . . . . . . . . 16  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
7978oveq1d 5796 . . . . . . . . . . . . . . 15  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8077, 79, 2ovmpog 5912 . . . . . . . . . . . . . 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 1217 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8274, 81eqtrd 2173 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
8382oveq1d 5796 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
8475nncnd 8757 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
8584, 38sqmuld 10466 . . . . . . . . . . 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 2173 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
8786oveq2d 5797 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( 2  x.  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2218 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  e.  CC )
8928nncnd 8757 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  CC )
9052, 88, 89mulassd 7812 . . . . . . . . 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 2176 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( 2  x.  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )  x.  (
( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2184 . . . . . . 7  |-  ( A  e.  NN  ->  (
2  x.  ( A ^ 2 ) )  =  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
93 df-ov 5784 . . . . . . 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 2190 . . . . . 6  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
)  =  ( 2  x.  ( A ^
2 ) ) )
95 f1ocnvfv 5687 . . . . . . 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 421 . . . . . 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 5432 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( 2nd `  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) >.
) )
99 op2ndg 6056 . . . . 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 2173 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  =  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) )
102101breq2d 3948 . 2  |-  ( A  e.  NN  ->  (
2  ||  ( 2nd `  ( `' F `  ( 2  x.  ( A ^ 2 ) ) ) )  <->  2  ||  ( ( ( 2nd `  ( `' F `  A ) )  x.  2 )  +  1 ) ) )
10320, 102mtbird 663 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 698    /\ w3a 963    = wceq 1332    e. wcel 1481   {crab 2421   <.cop 3534   class class class wbr 3936    X. cxp 4544   `'ccnv 4545   -->wf 5126   -1-1-onto->wf1o 5129   ` cfv 5130  (class class class)co 5781    e. cmpo 5783   1stc1st 6043   2ndc2nd 6044   CCcc 7641   1c1 7644    + caddc 7646    x. cmul 7648   NNcn 8743   2c2 8794   NN0cn0 9000   ZZcz 9077   ^cexp 10322    || cdvds 11527   Primecprime 11822
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-xor 1355  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-1o 6320  df-2o 6321  df-er 6436  df-en 6642  df-sup 6878  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fz 9821  df-fzo 9950  df-fl 10073  df-mod 10126  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-dvds 11528  df-gcd 11670  df-prm 11823
This theorem is referenced by:  sqne2sq  11889
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