Theorem 2sqpwodd 12369

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

Step	Hyp	Ref	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 ) )
3	1, 2	oddpwdc 12367	. . . . . . . 8 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5520	. . . . . . . 8 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5507	. . . . . . . 8 |- ( `' F : NN -1-1-onto-> ( J X. NN0 ) -> `' F : NN --> ( J X. NN0 ) )
6	3, 4, 5	mp2b 8	. . . . . . 7 |- `' F : NN --> ( J X. NN0 )
7	6	ffvelcdmi 5699	. . . . . 6 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6233	. . . . . 6 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( 2nd ` ( `' F ` A ) ) e. NN0 )
9	7, 8	syl 14	. . . . 5 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. NN0 )
10	9	nn0zd 9463	. . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9169	. . . . . 6 |- 2 e. NN
12	11	a1i 9	. . . . 5 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9464	. . . 4 |- ( A e. NN -> 2 e. ZZ )
14	10, 13	zmulcld 9471	. . 3 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ )
15		dvdsmul2 11996	. . . 4 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
16	10, 13, 15	syl2anc 411	. . 3 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
17		oddp1even 12058	. . . . 5 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ -> ( -. 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) <-> 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
18	17	biimprd 158	. . . 4 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ -> ( 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> -. 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
19	18	con2d 625	. . 3 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ -> ( 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> -. 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
20	14, 16, 19	sylc 62	. 2 |- ( A e. NN -> -. 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
21		xp1st 6232	. . . . . . . . . . 11 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( 1st ` ( `' F ` A ) ) e. J )
22	7, 21	syl 14	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. J )
23		breq2 4038	. . . . . . . . . . . . 13 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
24	23	notbid 668	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
25	24, 1	elrab2 2923	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
26	25	simplbi 274	. . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J -> ( 1st ` ( `' F ` A ) ) e. NN )
27	22, 26	syl 14	. . . . . . . . 9 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. NN )
28	27	nnsqcld 10803	. . . . . . . 8 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
29	25	simprbi 275	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) e. J -> -. 2 || ( 1st ` ( `' F ` A ) ) )
30	22, 29	syl 14	. . . . . . . . . 10 |- ( A e. NN -> -. 2 || ( 1st ` ( `' F ` A ) ) )
31		2prm 12320	. . . . . . . . . . 11 |- 2 e. Prime
32	27	nnzd 9464	. . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
33		euclemma 12339	. . . . . . . . . . . 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 758	. . . . . . . . . . . 12 |- ( ( 2 || ( 1st ` ( `' F ` A ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) )
35	33, 34	bitrdi 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 ) ) ) )
36	31, 32, 32, 35	mp3an2i 1353	. . . . . . . . . 10 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
37	30, 36	mtbird 674	. . . . . . . . 9 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
38	27	nncnd 9021	. . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
39	38	sqvald 10779	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
40	39	breq2d 4046	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
41	37, 40	mtbird 674	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
42		breq2 4038	. . . . . . . . . 10 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
43	42	notbid 668	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
44	43, 1	elrab2 2923	. . . . . . . 8 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
45	28, 41, 44	sylanbrc 417	. . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
46	12	nnnn0d 9319	. . . . . . . . 9 |- ( A e. NN -> 2 e. NN0 )
47	9, 46	nn0mulcld 9324	. . . . . . . 8 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
48		peano2nn0 9306	. . . . . . . 8 |- ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 -> ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) e. NN0 )
49	47, 48	syl 14	. . . . . . 7 |- ( A e. NN -> ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) e. NN0 )
50		opelxp 4694	. . . . . . 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 ) )
51	45, 49, 50	sylanbrc 417	. . . . . 6 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. e. ( J X. NN0 ) )
52	12	nncnd 9021	. . . . . . . . . . 11 |- ( A e. NN -> 2 e. CC )
53	52, 47	expp1d 10783	. . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) )
54	52, 47	expcld 10782	. . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. CC )
55	54, 52	mulcomd 8065	. . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) )
56	52, 46, 9	expmuld 10785	. . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
57	56	oveq2d 5941	. . . . . . . . . 10 |- ( A e. NN -> ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
58	53, 55, 57	3eqtrd 2233	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
59	58	oveq1d 5940	. . . . . . . 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 ) ) )
60	12, 49	nnexpcld 10804	. . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) e. NN )
61	60, 28	nnmulcld 9056	. . . . . . . . 9 |- ( A e. NN -> ( ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
62		oveq2 5933	. . . . . . . . . 10 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
63		oveq2 5933	. . . . . . . . . . 11 |- ( y = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> ( 2 ^ y ) = ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
64	63	oveq1d 5940	. . . . . . . . . 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 ) ) )
65	62, 64, 2	ovmpog 6061	. . . . . . . . 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 ) ) )
66	45, 49, 61, 65	syl3anc 1249	. . . . . . . 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 5828	. . . . . . . . . . . . . . . 16 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
68	3, 67	mpan 424	. . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
69		1st2nd2 6242	. . . . . . . . . . . . . . . . 17 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( `' F ` A ) = <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
70	7, 69	syl 14	. . . . . . . . . . . . . . . 16 |- ( A e. NN -> ( `' F ` A ) = <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
71	70	fveq2d 5565	. . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
72	68, 71	eqtr3d 2231	. . . . . . . . . . . . . 14 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
73		df-ov 5928	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
74	72, 73	eqtr4di 2247	. . . . . . . . . . . . 13 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
75	12, 9	nnexpcld 10804	. . . . . . . . . . . . . . 15 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
76	75, 27	nnmulcld 9056	. . . . . . . . . . . . . 14 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
77		oveq2 5933	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
78		oveq2 5933	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
79	78	oveq1d 5940	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
80	77, 79, 2	ovmpog 6061	. . . . . . . . . . . . . 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 ) ) ) )
81	22, 9, 76, 80	syl3anc 1249	. . . . . . . . . . . . 13 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
82	74, 81	eqtrd 2229	. . . . . . . . . . . 12 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
83	82	oveq1d 5940	. . . . . . . . . . 11 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
84	75	nncnd 9021	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
85	84, 38	sqmuld 10794	. . . . . . . . . . 11 |- ( A e. NN -> ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
86	83, 85	eqtrd 2229	. . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
87	86	oveq2d 5941	. . . . . . . . 9 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( 2 x. ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) ) )
88	56, 54	eqeltrrd 2274	. . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) e. CC )
89	28	nncnd 9021	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. CC )
90	52, 88, 89	mulassd 8067	. . . . . . . . 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 ) ) ) )
91	87, 90	eqtr4d 2232	. . . . . . . 8 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
92	59, 66, 91	3eqtr4rd 2240	. . . . . . 7 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
93		df-ov 5928	. . . . . . 7 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. )
94	92, 93	eqtr2di 2246	. . . . . 6 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( 2 x. ( A ^ 2 ) ) )
95		f1ocnvfv 5829	. . . . . . 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 ) >. ) )
96	3, 95	mpan 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 ) >. ) )
97	51, 94, 96	sylc 62	. . . . 5 |- ( A e. NN -> ( `' F ` ( 2 x. ( A ^ 2 ) ) ) = <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. )
98	97	fveq2d 5565	. . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) )
99		op2ndg 6218	. . . . 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 ) )
100	45, 49, 99	syl2anc 411	. . . 4 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
101	98, 100	eqtrd 2229	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
102	101	breq2d 4046	. 2 |- ( A e. NN -> ( 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) <-> 2 || ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
103	20, 102	mtbird 674	1 |- ( A e. NN -> -. 2 || ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   <.cop 3626   class class class wbr 4034   X. cxp 4662   `'ccnv 4663   -->wf 5255   -1-1-onto->wf1o 5258   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   1stc1st 6205   2ndc2nd 6206   CCcc 7894   1c1 7897   + caddc 7899   x. cmul 7901   NNcn 9007   2c2 9058   NN0cn0 9266   ZZcz 9343   ^cexp 10647   || cdvds 11969   Primecprime 12300

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-sup 7059  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-gcd 12146  df-prm 12301

This theorem is referenced by:  sqne2sq  12370