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Theorem 2sqpwodd 12108
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 12106 . . . . . . . 8 |- F : ( J X. NN0 ) -1-1-onto-> NN
4 f1ocnv 5445 . . . . . . . 8 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5 f1of 5432 . . . . . . . 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 5619 . . . . . 6 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8 xp2nd 6134 . . . . . 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 9311 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11 2nn 9018 . . . . . 6 |- 2 e. NN
1211a1i 9 . . . . 5 |- ( A e. NN -> 2 e. NN )
1312nnzd 9312 . . . 4 |- ( A e. NN -> 2 e. ZZ )
1410, 13zmulcld 9319 . . 3 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ )
15 dvdsmul2 11754 . . . 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 11813 . . . . 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 6133 . . . . . . . . . . 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 3986 . . . . . . . . . . . . 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 2885 . . . . . . . . . . 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 10609 . . . . . . . 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 12059 . . . . . . . . . . 11 |- 2 e. Prime
3227nnzd 9312 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
33 euclemma 12078 . . . . . . . . . . . 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, 34bitrdi 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 1332 . . . . . . . . . 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 8871 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
3938sqvald 10585 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
4039breq2d 3994 . . . . . . . . 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 3986 . . . . . . . . . 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 2885 . . . . . . . 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 9167 . . . . . . . . 9 |- ( A e. NN -> 2 e. NN0 )
479, 46nn0mulcld 9172 . . . . . . . 8 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
48 peano2nn0 9154 . . . . . . . 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 4634 . . . . . . 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 8871 . . . . . . . . . . 11 |- ( A e. NN -> 2 e. CC )
5352, 47expp1d 10589 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) )
5452, 47expcld 10588 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. CC )
5554, 52mulcomd 7920 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) )
5652, 46, 9expmuld 10591 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
5756oveq2d 5858 . . . . . . . . . 10 |- ( A e. NN -> ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2202 . . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5958oveq1d 5857 . . . . . . . 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 10610 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) e. NN )
6160, 28nnmulcld 8906 . . . . . . . . 9 |- ( A e. NN -> ( ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
62 oveq2 5850 . . . . . . . . . 10 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
63 oveq2 5850 . . . . . . . . . . 11 |- ( y = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> ( 2 ^ y ) = ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
6463oveq1d 5857 . . . . . . . . . 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 5976 . . . . . . . . 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 1228 . . . . . . . 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 5746 . . . . . . . . . . . . . . . 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 6143 . . . . . . . . . . . . . . . . 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 5490 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
7268, 71eqtr3d 2200 . . . . . . . . . . . . . 14 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
73 df-ov 5845 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
7472, 73eqtr4di 2217 . . . . . . . . . . . . 13 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
7512, 9nnexpcld 10610 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
7675, 27nnmulcld 8906 . . . . . . . . . . . . . 14 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
77 oveq2 5850 . . . . . . . . . . . . . . 15 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
78 oveq2 5850 . . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
7978oveq1d 5857 . . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8077, 79, 2ovmpog 5976 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 13 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8274, 81eqtrd 2198 . . . . . . . . . . . 12 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8382oveq1d 5857 . . . . . . . . . . 11 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
8475nncnd 8871 . . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
8584, 38sqmuld 10600 . . . . . . . . . . 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 2198 . . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
8786oveq2d 5858 . . . . . . . . 9 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( 2 x. ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2244 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) e. CC )
8928nncnd 8871 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. CC )
9052, 88, 89mulassd 7922 . . . . . . . . 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 2201 . . . . . . . 8 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2209 . . . . . . 7 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
93 df-ov 5845 . . . . . . 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 2216 . . . . . 6 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( 2 x. ( A ^ 2 ) ) )
95 f1ocnvfv 5747 . . . . . . 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 5490 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) )
99 op2ndg 6119 . . . . 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 2198 . . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
102101breq2d 3994 . 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 968   = wceq 1343   e. wcel 2136   {crab 2448   <.cop 3579   class class class wbr 3982   X. cxp 4602   `'ccnv 4603   -->wf 5184   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   ^cexp 10454   || cdvds 11727   Primecprime 12039
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876  df-prm 12040
This theorem is referenced by:  sqne2sq  12109
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