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Theorem sqpweven 11764
Description: The greatest power of two dividing the square of an integer is an even 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
sqpweven  |-  ( A  e.  NN  ->  2  ||  ( 2nd `  ( `' F `  ( 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 sqpweven
StepHypRef Expression
1 oddpwdc.j . . . . . . . 8  |-  J  =  { z  e.  NN  |  -.  2  ||  z }
2 oddpwdc.f . . . . . . . 8  |-  F  =  ( x  e.  J ,  y  e.  NN0  |->  ( ( 2 ^ y )  x.  x
) )
31, 2oddpwdc 11763 . . . . . . 7  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5348 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5335 . . . . . . 7  |-  ( `' F : NN -1-1-onto-> ( J  X.  NN0 )  ->  `' F : NN
--> ( J  X.  NN0 ) )
63, 4, 5mp2b 8 . . . . . 6  |-  `' F : NN --> ( J  X.  NN0 )
76ffvelrni 5522 . . . . 5  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 6032 . . . . 5  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( 2nd `  ( `' F `  A ) )  e. 
NN0 )
97, 8syl 14 . . . 4  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e. 
NN0 )
109nn0zd 9129 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 8839 . . . . 5  |-  2  e.  NN
1211a1i 9 . . . 4  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 9130 . . 3  |-  ( A  e.  NN  ->  2  e.  ZZ )
14 dvdsmul2 11428 . . 3  |-  ( ( ( 2nd `  ( `' F `  A ) )  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
1510, 13, 14syl2anc 408 . 2  |-  ( A  e.  NN  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
16 xp1st 6031 . . . . . . . . . 10  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( 1st `  ( `' F `  A ) )  e.  J )
177, 16syl 14 . . . . . . . . 9  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  J )
18 breq2 3903 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
1918notbid 641 . . . . . . . . . . 11  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2019, 1elrab2 2816 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  <->  ( ( 1st `  ( `' F `  A ) )  e.  NN  /\  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2120simplbi 272 . . . . . . . . 9  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  -> 
( 1st `  ( `' F `  A ) )  e.  NN )
2217, 21syl 14 . . . . . . . 8  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  NN )
2322nnsqcld 10400 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN )
2420simprbi 273 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
2517, 24syl 14 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
26 2prm 11720 . . . . . . . . . 10  |-  2  e.  Prime
2722nnzd 9130 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
28 euclemma 11736 . . . . . . . . . . 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 ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) ) ) )
29 oridm 731 . . . . . . . . . . 11  |-  ( ( 2  ||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) )
3028, 29syl6bb 195 . . . . . . . . . 10  |-  ( ( 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 ) ) ) )
3126, 27, 27, 30mp3an2i 1305 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3225, 31mtbird 647 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3322nncnd 8698 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3433sqvald 10376 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3534breq2d 3911 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
3632, 35mtbird 647 . . . . . . 7  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
37 breq2 3903 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
3837notbid 641 . . . . . . . 8  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
3938, 1elrab2 2816 . . . . . . 7  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  <->  ( (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN  /\  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
4023, 36, 39sylanbrc 413 . . . . . 6  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J )
4112nnnn0d 8988 . . . . . . 7  |-  ( A  e.  NN  ->  2  e.  NN0 )
429, 41nn0mulcld 8993 . . . . . 6  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
43 opelxp 4539 . . . . . 6  |-  ( <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >.  e.  ( J  X.  NN0 )  <->  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  /\  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 ) )
4440, 42, 43sylanbrc 413 . . . . 5  |-  ( A  e.  NN  ->  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >.  e.  ( J  X.  NN0 )
)
4512nncnd 8698 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  CC )
4645, 41, 9expmuld 10382 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
4746oveq1d 5757 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
4812, 42nnexpcld 10401 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  NN )
4948, 23nnmulcld 8733 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) )  e.  NN )
50 oveq2 5750 . . . . . . . . 9  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
51 oveq2 5750 . . . . . . . . . 10  |-  ( y  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  ->  (
2 ^ y )  =  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
5251oveq1d 5757 . . . . . . . . 9  |-  ( y  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  ->  (
( 2 ^ y
)  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
5350, 52, 2ovmpog 5873 . . . . . . . 8  |-  ( ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  /\  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0  /\  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) )  e.  NN )  ->  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
5440, 42, 49, 53syl3anc 1201 . . . . . . 7  |-  ( A  e.  NN  ->  (
( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
55 f1ocnvfv2 5647 . . . . . . . . . . . . 13  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  A  e.  NN )  ->  ( F `  ( `' F `  A ) )  =  A )
563, 55mpan 420 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  A )
57 1st2nd2 6041 . . . . . . . . . . . . . 14  |-  ( ( `' F `  A )  e.  ( J  X.  NN0 )  ->  ( `' F `  A )  =  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
587, 57syl 14 . . . . . . . . . . . . 13  |-  ( A  e.  NN  ->  ( `' F `  A )  =  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
5958fveq2d 5393 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
6056, 59eqtr3d 2152 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
61 df-ov 5745 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
6260, 61syl6eqr 2168 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
6312, 9nnexpcld 10401 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
6463, 22nnmulcld 8733 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
65 oveq2 5750 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
66 oveq2 5750 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
6766oveq1d 5757 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
6865, 67, 2ovmpog 5873 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) ) )
6917, 9, 64, 68syl3anc 1201 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
7062, 69eqtrd 2150 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
7170oveq1d 5757 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
7263nncnd 8698 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
7372, 33sqmuld 10391 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
7471, 73eqtrd 2150 . . . . . . 7  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
7547, 54, 743eqtr4rd 2161 . . . . . 6  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 1st `  ( `' F `  A ) ) ^
2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
76 df-ov 5745 . . . . . 6  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )
7775, 76syl6req 2167 . . . . 5  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )  =  ( A ^
2 ) )
78 f1ocnvfv 5648 . . . . . 6  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >.  e.  ( J  X.  NN0 )
)  ->  ( ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )  =  ( A ^
2 )  ->  ( `' F `  ( A ^ 2 ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. ) )
793, 78mpan 420 . . . . 5  |-  ( <.
( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >.  e.  ( J  X.  NN0 )  ->  ( ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )  =  ( A ^
2 )  ->  ( `' F `  ( A ^ 2 ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. ) )
8044, 77, 79sylc 62 . . . 4  |-  ( A  e.  NN  ->  ( `' F `  ( A ^ 2 ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. )
8180fveq2d 5393 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( A ^ 2 ) ) )  =  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )
)
82 op2ndg 6017 . . . 4  |-  ( ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  /\  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )  -> 
( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
8340, 42, 82syl2anc 408 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. )  =  (
( 2nd `  ( `' F `  A ) )  x.  2 ) )
8481, 83eqtrd 2150 . 2  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( A ^ 2 ) ) )  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
8515, 84breqtrrd 3926 1  |-  ( A  e.  NN  ->  2  ||  ( 2nd `  ( `' F `  ( A ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   {crab 2397   <.cop 3500   class class class wbr 3899    X. cxp 4507   `'ccnv 4508   -->wf 5089   -1-1-onto->wf1o 5092   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   1stc1st 6004   2ndc2nd 6005    x. cmul 7593   NNcn 8684   2c2 8735   NN0cn0 8935   ZZcz 9012   ^cexp 10247    || cdvds 11405   Primecprime 11700
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-stab 801  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-1o 6281  df-2o 6282  df-er 6397  df-en 6603  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-q 9368  df-rp 9398  df-fz 9746  df-fzo 9875  df-fl 9998  df-mod 10051  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-dvds 11406  df-gcd 11548  df-prm 11701
This theorem is referenced by:  sqne2sq  11766
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