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Theorem 2sqpwodd 12176
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 12174 . . . . . . . 8 |- F : ( J X. NN0 ) -1-1-onto-> NN
4 f1ocnv 5475 . . . . . . . 8 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5 f1of 5462 . . . . . . . 8 |- ( `' F : NN -1-1-onto-> ( J X. NN0 ) -> `' F : NN --> ( J X. NN0 ) )
63, 4, 5mp2b 8 . . . . . . 7 |- `' F : NN --> ( J X. NN0 )
76ffvelcdmi 5651 . . . . . 6 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8 xp2nd 6167 . . . . . 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 9373 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11 2nn 9080 . . . . . 6 |- 2 e. NN
1211a1i 9 . . . . 5 |- ( A e. NN -> 2 e. NN )
1312nnzd 9374 . . . 4 |- ( A e. NN -> 2 e. ZZ )
1410, 13zmulcld 9381 . . 3 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ )
15 dvdsmul2 11821 . . . 4 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
1610, 13, 15syl2anc 411 . . 3 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
17 oddp1even 11881 . . . . 5 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ -> ( -. 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) <-> 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
1817biimprd 158 . . . 4 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ -> ( 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> -. 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
1918con2d 624 . . 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 6166 . . . . . . . . . . 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 4008 . . . . . . . . . . . . 13 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
2423notbid 667 . . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
2524, 1elrab2 2897 . . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
2625simplbi 274 . . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J -> ( 1st ` ( `' F ` A ) ) e. NN )
2722, 26syl 14 . . . . . . . . 9 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. NN )
2827nnsqcld 10675 . . . . . . . 8 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
2925simprbi 275 . . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J -> -. 2 || ( 1st ` ( `' F ` A ) ) )
3022, 29syl 14 . . . . . . . . . 10 |- ( A e. NN -> -. 2 || ( 1st ` ( `' F ` A ) ) )
31 2prm 12127 . . . . . . . . . . 11 |- 2 e. Prime
3227nnzd 9374 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
33 euclemma 12146 . . . . . . . . . . . 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 757 . . . . . . . . . . . 12 |- ( ( 2 || ( 1st ` ( `' F ` A ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) )
3533, 34bitrdi 196 . . . . . . . . . . 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 1342 . . . . . . . . . 10 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
3730, 36mtbird 673 . . . . . . . . 9 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
3827nncnd 8933 . . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
3938sqvald 10651 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
4039breq2d 4016 . . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
4137, 40mtbird 673 . . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
42 breq2 4008 . . . . . . . . . 10 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4342notbid 667 . . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4443, 1elrab2 2897 . . . . . . . 8 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
4528, 41, 44sylanbrc 417 . . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
4612nnnn0d 9229 . . . . . . . . 9 |- ( A e. NN -> 2 e. NN0 )
479, 46nn0mulcld 9234 . . . . . . . 8 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
48 peano2nn0 9216 . . . . . . . 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 4657 . . . . . . 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 417 . . . . . 6 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. e. ( J X. NN0 ) )
5212nncnd 8933 . . . . . . . . . . 11 |- ( A e. NN -> 2 e. CC )
5352, 47expp1d 10655 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) )
5452, 47expcld 10654 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. CC )
5554, 52mulcomd 7979 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) )
5652, 46, 9expmuld 10657 . . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
5756oveq2d 5891 . . . . . . . . . 10 |- ( A e. NN -> ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5853, 55, 573eqtrd 2214 . . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
5958oveq1d 5890 . . . . . . . 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 10676 . . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) e. NN )
6160, 28nnmulcld 8968 . . . . . . . . 9 |- ( A e. NN -> ( ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
62 oveq2 5883 . . . . . . . . . 10 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
63 oveq2 5883 . . . . . . . . . . 11 |- ( y = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> ( 2 ^ y ) = ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
6463oveq1d 5890 . . . . . . . . . 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 6009 . . . . . . . . 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 1238 . . . . . . . 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 5779 . . . . . . . . . . . . . . . 16 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
683, 67mpan 424 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
69 1st2nd2 6176 . . . . . . . . . . . . . . . . 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 5520 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
7268, 71eqtr3d 2212 . . . . . . . . . . . . . 14 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
73 df-ov 5878 . . . . . . . . . . . . . 14 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
7472, 73eqtr4di 2228 . . . . . . . . . . . . 13 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
7512, 9nnexpcld 10676 . . . . . . . . . . . . . . 15 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
7675, 27nnmulcld 8968 . . . . . . . . . . . . . 14 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
77 oveq2 5883 . . . . . . . . . . . . . . 15 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
78 oveq2 5883 . . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
7978oveq1d 5890 . . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8077, 79, 2ovmpog 6009 . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . 13 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8274, 81eqtrd 2210 . . . . . . . . . . . 12 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
8382oveq1d 5890 . . . . . . . . . . 11 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
8475nncnd 8933 . . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
8584, 38sqmuld 10666 . . . . . . . . . . 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 2210 . . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
8786oveq2d 5891 . . . . . . . . 9 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( 2 x. ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) ) )
8856, 54eqeltrrd 2255 . . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) e. CC )
8928nncnd 8933 . . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. CC )
9052, 88, 89mulassd 7981 . . . . . . . . 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 2213 . . . . . . . 8 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
9259, 66, 913eqtr4rd 2221 . . . . . . 7 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
93 df-ov 5878 . . . . . . 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 2227 . . . . . 6 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( 2 x. ( A ^ 2 ) ) )
95 f1ocnvfv 5780 . . . . . . 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 424 . . . . . 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 5520 . . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) )
99 op2ndg 6152 . . . . 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 411 . . . 4 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
10198, 100eqtrd 2210 . . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
102101breq2d 4016 . 2 |- ( A e. NN -> ( 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) <-> 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
10320, 102mtbird 673 1 |- ( A e. NN -> -. 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   {crab 2459   <.cop 3596   class class class wbr 4004   X. cxp 4625   `'ccnv 4626   -->wf 5213   -1-1-onto->wf1o 5216   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   1stc1st 6139   2ndc2nd 6140   CCcc 7809   1c1 7812   + caddc 7814   x. cmul 7816   NNcn 8919   2c2 8970   NN0cn0 9176   ZZcz 9253   ^cexp 10519   || cdvds 11794   Primecprime 12107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-1o 6417  df-2o 6418  df-er 6535  df-en 6741  df-sup 6983  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-fl 10270  df-mod 10323  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-dvds 11795  df-gcd 11944  df-prm 12108
This theorem is referenced by:  sqne2sq  12177
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