Theorem 2sqpwodd 12314

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 12312	. . . . . . . 8 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5513	. . . . . . . 8 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5500	. . . . . . . 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 5692	. . . . . 6 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6219	. . . . . 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 9437	. . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9143	. . . . . 6 |- 2 e. NN
12	11	a1i 9	. . . . 5 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9438	. . . 4 |- ( A e. NN -> 2 e. ZZ )
14	10, 13	zmulcld 9445	. . 3 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. ZZ )
15		dvdsmul2 11957	. . . 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 12017	. . . . 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 6218	. . . . . . . . . . 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 4033	. . . . . . . . . . . . 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 2919	. . . . . . . . . . 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 10765	. . . . . . . 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 12265	. . . . . . . . . . 11 |- 2 e. Prime
32	27	nnzd 9438	. . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
33		euclemma 12284	. . . . . . . . . . . 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 8996	. . . . . . . . . . 11 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
39	38	sqvald 10741	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
40	39	breq2d 4041	. . . . . . . . 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 4033	. . . . . . . . . 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 2919	. . . . . . . 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 9293	. . . . . . . . 9 |- ( A e. NN -> 2 e. NN0 )
47	9, 46	nn0mulcld 9298	. . . . . . . 8 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
48		peano2nn0 9280	. . . . . . . 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 4689	. . . . . . 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 8996	. . . . . . . . . . 11 |- ( A e. NN -> 2 e. CC )
53	52, 47	expp1d 10745	. . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. 2 ) )
54	52, 47	expcld 10744	. . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. CC )
55	54, 52	mulcomd 8041	. . . . . . . . . 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 10747	. . . . . . . . . . 11 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
57	56	oveq2d 5934	. . . . . . . . . 10 |- ( A e. NN -> ( 2 x. ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
58	53, 55, 57	3eqtrd 2230	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) = ( 2 x. ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) ) )
59	58	oveq1d 5933	. . . . . . . 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 10766	. . . . . . . . . 10 |- ( A e. NN -> ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) e. NN )
61	60, 28	nnmulcld 9031	. . . . . . . . 9 |- ( A e. NN -> ( ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
62		oveq2 5926	. . . . . . . . . 10 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
63		oveq2 5926	. . . . . . . . . . 11 |- ( y = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) -> ( 2 ^ y ) = ( 2 ^ ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
64	63	oveq1d 5933	. . . . . . . . . 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 6053	. . . . . . . . 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 5821	. . . . . . . . . . . . . . . 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 6228	. . . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 15 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
72	68, 71	eqtr3d 2228	. . . . . . . . . . . . . 14 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
73		df-ov 5921	. . . . . . . . . . . . . 14 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
74	72, 73	eqtr4di 2244	. . . . . . . . . . . . 13 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
75	12, 9	nnexpcld 10766	. . . . . . . . . . . . . . 15 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
76	75, 27	nnmulcld 9031	. . . . . . . . . . . . . 14 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
77		oveq2 5926	. . . . . . . . . . . . . . 15 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
78		oveq2 5926	. . . . . . . . . . . . . . . 16 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
79	78	oveq1d 5933	. . . . . . . . . . . . . . 15 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
80	77, 79, 2	ovmpog 6053	. . . . . . . . . . . . . 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 2226	. . . . . . . . . . . 12 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
83	82	oveq1d 5933	. . . . . . . . . . 11 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
84	75	nncnd 8996	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
85	84, 38	sqmuld 10756	. . . . . . . . . . 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 2226	. . . . . . . . . 10 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
87	86	oveq2d 5934	. . . . . . . . 9 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( 2 x. ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) ) )
88	56, 54	eqeltrrd 2271	. . . . . . . . . 10 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) e. CC )
89	28	nncnd 8996	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. CC )
90	52, 88, 89	mulassd 8043	. . . . . . . . 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 2229	. . . . . . . 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 2237	. . . . . . 7 |- ( A e. NN -> ( 2 x. ( A ^ 2 ) ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) ) )
93		df-ov 5921	. . . . . . 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 2243	. . . . . 6 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) = ( 2 x. ( A ^ 2 ) ) )
95		f1ocnvfv 5822	. . . . . . 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 5558	. . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) >. ) )
99		op2ndg 6204	. . . . 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 2226	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( 2 x. ( A ^ 2 ) ) ) ) = ( ( ( 2nd ` ( `' F ` A ) ) x. 2 ) + 1 ) )
102	101	breq2d 4041	. 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 2164   {crab 2476   <.cop 3621   class class class wbr 4029   X. cxp 4657   `'ccnv 4658   -->wf 5250   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1stc1st 6191   2ndc2nd 6192   CCcc 7870   1c1 7873   + caddc 7875   x. cmul 7877   NNcn 8982   2c2 9033   NN0cn0 9240   ZZcz 9317   ^cexp 10609   || cdvds 11930   Primecprime 12245

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246

This theorem is referenced by:  sqne2sq  12315