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Theorem sqpweven 10778
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 10777 . . . . . . 7  |-  F :
( J  X.  NN0 )
-1-1-onto-> NN
4 f1ocnv 5191 . . . . . . 7  |-  ( F : ( J  X.  NN0 ) -1-1-onto-> NN  ->  `' F : NN -1-1-onto-> ( J  X.  NN0 ) )
5 f1of 5178 . . . . . . 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 5354 . . . . 5  |-  ( A  e.  NN  ->  ( `' F `  A )  e.  ( J  X.  NN0 ) )
8 xp2nd 5845 . . . . 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 8618 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  A ) )  e.  ZZ )
11 2nn 8330 . . . . 5  |-  2  e.  NN
1211a1i 9 . . . 4  |-  ( A  e.  NN  ->  2  e.  NN )
1312nnzd 8619 . . 3  |-  ( A  e.  NN  ->  2  e.  ZZ )
14 dvdsmul2 10444 . . 3  |-  ( ( ( 2nd `  ( `' F `  A ) )  e.  ZZ  /\  2  e.  ZZ )  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
1510, 13, 14syl2anc 403 . 2  |-  ( A  e.  NN  ->  2  ||  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
16 xp1st 5844 . . . . . . . . . 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 3809 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( 2 
||  z  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
1918notbid 625 . . . . . . . . . . 11  |-  ( z  =  ( 1st `  ( `' F `  A ) )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2019, 1elrab2 2760 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  <->  ( ( 1st `  ( `' F `  A ) )  e.  NN  /\  -.  2  ||  ( 1st `  ( `' F `  A ) ) ) )
2120simplbi 268 . . . . . . . . 9  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  -> 
( 1st `  ( `' F `  A ) )  e.  NN )
2217, 21syl 14 . . . . . . . 8  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  NN )
2322nnsqcld 9793 . . . . . . 7  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN )
2420simprbi 269 . . . . . . . . . 10  |-  ( ( 1st `  ( `' F `  A ) )  e.  J  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
2517, 24syl 14 . . . . . . . . 9  |-  ( A  e.  NN  ->  -.  2  ||  ( 1st `  ( `' F `  A ) ) )
26 2prm 10734 . . . . . . . . . 10  |-  2  e.  Prime
2722nnzd 8619 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  ZZ )
28 euclemma 10750 . . . . . . . . . . 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 707 . . . . . . . . . . 11  |-  ( ( 2  ||  ( 1st `  ( `' F `  A ) )  \/  2  ||  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) )
3028, 29syl6bb 194 . . . . . . . . . 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 1274 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) )  <->  2  ||  ( 1st `  ( `' F `  A ) ) ) )
3225, 31mtbird 631 . . . . . . . 8  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3322nncnd 8190 . . . . . . . . . 10  |-  ( A  e.  NN  ->  ( 1st `  ( `' F `  A ) )  e.  CC )
3433sqvald 9769 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  =  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
3534breq2d 3817 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 )  <->  2  ||  ( ( 1st `  ( `' F `  A ) )  x.  ( 1st `  ( `' F `  A ) ) ) ) )
3632, 35mtbird 631 . . . . . . 7  |-  ( A  e.  NN  ->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) )
37 breq2 3809 . . . . . . . . 9  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
2  ||  z  <->  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
3837notbid 625 . . . . . . . 8  |-  ( z  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  ( -.  2  ||  z  <->  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
3938, 1elrab2 2760 . . . . . . 7  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J  <->  ( (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  NN  /\  -.  2  ||  ( ( 1st `  ( `' F `  A ) ) ^
2 ) ) )
4023, 36, 39sylanbrc 408 . . . . . 6  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) ^ 2 )  e.  J )
4112nnnn0d 8478 . . . . . . 7  |-  ( A  e.  NN  ->  2  e.  NN0 )
429, 41nn0mulcld 8483 . . . . . 6  |-  ( A  e.  NN  ->  (
( 2nd `  ( `' F `  A ) )  x.  2 )  e.  NN0 )
43 opelxp 4420 . . . . . 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 408 . . . . 5  |-  ( A  e.  NN  ->  <. (
( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >.  e.  ( J  X.  NN0 )
)
4512nncnd 8190 . . . . . . . . 9  |-  ( A  e.  NN  ->  2  e.  CC )
4645, 41, 9expmuld 9775 . . . . . . . 8  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 ) )
4746oveq1d 5579 . . . . . . 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 9794 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  e.  NN )
4948, 23nnmulcld 8224 . . . . . . . 8  |-  ( A  e.  NN  ->  (
( 2 ^ (
( 2nd `  ( `' F `  A ) )  x.  2 ) )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) )  e.  NN )
50 oveq2 5572 . . . . . . . . 9  |-  ( x  =  ( ( 1st `  ( `' F `  A ) ) ^
2 )  ->  (
( 2 ^ y
)  x.  x )  =  ( ( 2 ^ y )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
51 oveq2 5572 . . . . . . . . . 10  |-  ( y  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 )  ->  (
2 ^ y )  =  ( 2 ^ ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
5251oveq1d 5579 . . . . . . . . 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, 2ovmpt2g 5687 . . . . . . . 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 1170 . . . . . . 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 5470 . . . . . . . . . . . . 13  |-  ( ( F : ( J  X.  NN0 ) -1-1-onto-> NN  /\  A  e.  NN )  ->  ( F `  ( `' F `  A ) )  =  A )
563, 55mpan 415 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  A )
57 1st2nd2 5853 . . . . . . . . . . . . . 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 5234 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  ( F `  ( `' F `  A )
)  =  ( F `
 <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
6056, 59eqtr3d 2117 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  A  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
) )
61 df-ov 5567 . . . . . . . . . . 11  |-  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( F `  <. ( 1st `  ( `' F `  A ) ) ,  ( 2nd `  ( `' F `  A ) ) >.
)
6260, 61syl6eqr 2133 . . . . . . . . . 10  |-  ( A  e.  NN  ->  A  =  ( ( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) ) )
6312, 9nnexpcld 9794 . . . . . . . . . . . 12  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  NN )
6463, 22nnmulcld 8224 . . . . . . . . . . 11  |-  ( A  e.  NN  ->  (
( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) )  e.  NN )
65 oveq2 5572 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  x )  =  ( ( 2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) ) )
66 oveq2 5572 . . . . . . . . . . . . 13  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( 2 ^ y )  =  ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) )
6766oveq1d 5579 . . . . . . . . . . . 12  |-  ( y  =  ( 2nd `  ( `' F `  A ) )  ->  ( (
2 ^ y )  x.  ( 1st `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
6865, 67, 2ovmpt2g 5687 . . . . . . . . . . 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 1170 . . . . . . . . . 10  |-  ( A  e.  NN  ->  (
( 1st `  ( `' F `  A ) ) F ( 2nd `  ( `' F `  A ) ) )  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
7062, 69eqtrd 2115 . . . . . . . . 9  |-  ( A  e.  NN  ->  A  =  ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) )
7170oveq1d 5579 . . . . . . . 8  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) )  x.  ( 1st `  ( `' F `  A ) ) ) ^ 2 ) )
7263nncnd 8190 . . . . . . . . 9  |-  ( A  e.  NN  ->  (
2 ^ ( 2nd `  ( `' F `  A ) ) )  e.  CC )
7372, 33sqmuld 9784 . . . . . . . 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 2115 . . . . . . 7  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 2 ^ ( 2nd `  ( `' F `  A ) ) ) ^ 2 )  x.  ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ) )
7547, 54, 743eqtr4rd 2126 . . . . . 6  |-  ( A  e.  NN  ->  ( A ^ 2 )  =  ( ( ( 1st `  ( `' F `  A ) ) ^
2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) ) )
76 df-ov 5567 . . . . . 6  |-  ( ( ( 1st `  ( `' F `  A ) ) ^ 2 ) F ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )  =  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )
7775, 76syl6req 2132 . . . . 5  |-  ( A  e.  NN  ->  ( F `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )  =  ( A ^
2 ) )
78 f1ocnvfv 5471 . . . . . 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 415 . . . . 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 61 . . . 4  |-  ( A  e.  NN  ->  ( `' F `  ( A ^ 2 ) )  =  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. )
8180fveq2d 5234 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( A ^ 2 ) ) )  =  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^ 2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) >. )
)
82 op2ndg 5830 . . . 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 403 . . 3  |-  ( A  e.  NN  ->  ( 2nd `  <. ( ( 1st `  ( `' F `  A ) ) ^
2 ) ,  ( ( 2nd `  ( `' F `  A ) )  x.  2 )
>. )  =  (
( 2nd `  ( `' F `  A ) )  x.  2 ) )
8481, 83eqtrd 2115 . 2  |-  ( A  e.  NN  ->  ( 2nd `  ( `' F `  ( A ^ 2 ) ) )  =  ( ( 2nd `  ( `' F `  A ) )  x.  2 ) )
8515, 84breqtrrd 3831 1  |-  ( A  e.  NN  ->  2  ||  ( 2nd `  ( `' F `  ( 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    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-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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