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Theorem 2sqpwodd 11693
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 11691 . . . . . . . 8 |- F : ( J X. NN0 ) -1-1-onto-> NN
4 f1ocnv 5334 . . . . . . . 8 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5 f1of 5321 . . . . . . . 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 5506 . . . . . 6 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8 xp2nd 6016 . . . . . 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 9069 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11 2nn 8779 . . . . . 6 |- 2 e. NN
1211a1i 9 . . . . 5 |- ( A e. NN -> 2 e. NN )
1312nnzd 9070 . . . 4 |- ( A e. NN -> 2 e. ZZ )
1410, 13zmulcld 9077 . . 3 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ )
15 dvdsmul2 11358 . . . 4 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
1610, 13, 15syl2anc 406 . . 3 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
17 oddp1even 11415 . . . . 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 596 . . 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 6015 . . . . . . . . . . 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 3897 . . . . . . . . . . . . 13 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
2423notbid 639 . . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
2524, 1elrab2 2810 . . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
2625simplbi 270 . . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J -> ( 1st ` ( `' F ` A ) ) e. NN )
2722, 26syl 14 . . . . . . . . 9 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. NN )
2827nnsqcld 10332 . . . . . . . 8 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
2925simprbi 271 . . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J -> -. 2 || ( 1st ` ( `' F ` A ) ) )
3022, 29syl 14 . . . . . . . . . 10 |- ( A e. NN -> -. 2 || ( 1st ` ( `' F ` A ) ) )
31 2prm 11648 . . . . . . . . . . 11 |- 2 e. Prime
3227nnzd 9070 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
33 euclemma 11664 . . . . . . . . . . . 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 729 . . . . . . . . . . . 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 1301 . . . . . . . . . 10 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
3730, 36mtbird 645 . . . . . . . . 9 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
3827nncnd 8638 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
3938sqvald 10308 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
4039breq2d 3905 . . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
4137, 40mtbird 645 . . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
42 breq2 3897 . . . . . . . . . 10 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4342notbid 639 . . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4443, 1elrab2 2810 . . . . . . . 8 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4528, 41, 44sylanbrc 411 . . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
4612nnnn0d 8928 . . . . . . . . 9 |- ( A e. NN -> 2 e. NN0 )
479, 46nn0mulcld 8933 . . . . . . . 8 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
48 peano2nn0 8915 . . . . . . . 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 4527 . . . . . . 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 411 . . . . . 6 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. e. ( J X. NN0 ) )
5212nncnd 8638 . . . . . . . . . . 11 |- ( A e. NN -> 2 e. CC )
5352, 47expp1d 10312 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) )
5452, 47expcld 10311 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. CC )
5554, 52mulcomd 7705 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) )
5652, 46, 9expmuld 10314 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
5756oveq2d 5742 . . . . . . . . . 10 |- ( A e. NN -> ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2149 . . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5958oveq1d 5741 . . . . . . . 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 10333 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) e. NN )
6160, 28nnmulcld 8673 . . . . . . . . 9 |- ( A e. NN -> ( ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
62 oveq2 5734 . . . . . . . . . 10 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
63 oveq2 5734 . . . . . . . . . . 11 |- ( y = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> ( 2 ^ y ) = ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
6463oveq1d 5741 . . . . . . . . . 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 5857 . . . . . . . . 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 1197 . . . . . . . 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 5631 . . . . . . . . . . . . . . . 16 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
683, 67mpan 418 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
69 1st2nd2 6025 . . . . . . . . . . . . . . . . 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 5377 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
7268, 71eqtr3d 2147 . . . . . . . . . . . . . 14 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
73 df-ov 5729 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
7472, 73syl6eqr 2163 . . . . . . . . . . . . 13 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
7512, 9nnexpcld 10333 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
7675, 27nnmulcld 8673 . . . . . . . . . . . . . 14 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
77 oveq2 5734 . . . . . . . . . . . . . . 15 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
78 oveq2 5734 . . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
7978oveq1d 5741 . . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8077, 79, 2ovmpog 5857 . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . 13 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8274, 81eqtrd 2145 . . . . . . . . . . . 12 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8382oveq1d 5741 . . . . . . . . . . 11 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
8475nncnd 8638 . . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
8584, 38sqmuld 10323 . . . . . . . . . . 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 2145 . . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
8786oveq2d 5742 . . . . . . . . 9 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( 2 x. ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2190 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) e. CC )
8928nncnd 8638 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. CC )
9052, 88, 89mulassd 7707 . . . . . . . . 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 2148 . . . . . . . 8 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2156 . . . . . . 7 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
93 df-ov 5729 . . . . . . 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 2162 . . . . . 6 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( 2 x. ( A ^ 2 ) ) )
95 f1ocnvfv 5632 . . . . . . 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 418 . . . . . 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 5377 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) )
99 op2ndg 6001 . . . . 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 406 . . . 4 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
10198, 100eqtrd 2145 . . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
102101breq2d 3905 . 2 |- ( A e. NN -> ( 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) <-> 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
10320, 102mtbird 645 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 680   /\ w3a 943   = wceq 1312   e. wcel 1461   {crab 2392   <.cop 3494   class class class wbr 3893   X. cxp 4495   `'ccnv 4496   -->wf 5075   -1-1-onto->wf1o 5078   ` cfv 5079  (class class class)co 5726   e. cmpo 5728   1stc1st 5988   2ndc2nd 5989   CCcc 7539   1c1 7542   + caddc 7544   x. cmul 7546   NNcn 8624   2c2 8675   NN0cn0 8875   ZZcz 8952   ^cexp 10179   || cdvds 11335   Primecprime 11628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657  ax-arch 7658  ax-caucvg 7659
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-xor 1335  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-if 3439  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-ilim 4249  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-frec 6240  df-1o 6265  df-2o 6266  df-er 6381  df-en 6587  df-sup 6821  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-inn 8625  df-2 8683  df-3 8684  df-4 8685  df-n0 8876  df-z 8953  df-uz 9223  df-q 9308  df-rp 9338  df-fz 9678  df-fzo 9807  df-fl 9930  df-mod 9983  df-seqfrec 10106  df-exp 10180  df-cj 10501  df-re 10502  df-im 10503  df-rsqrt 10656  df-abs 10657  df-dvds 11336  df-gcd 11478  df-prm 11629
This theorem is referenced by:  sqne2sq  11694
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