Theorem sqpweven 12343

Description: The greatest power of two dividing the square of an integer is an even 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
sqpweven	|- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( 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 sqpweven

Step	Hyp	Ref	Expression
1		oddpwdc.j	. . . . . . . 8 |- J = { z e. NN | -. 2 || z }
2		oddpwdc.f	. . . . . . . 8 |- F = ( x e. J , y e. NN0 |-> ( ( 2 ^ y ) x. x ) )
3	1, 2	oddpwdc 12342	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5517	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5504	. . . . . . 7 |- ( `' F : NN -1-1-onto-> ( J X. NN0 ) -> `' F : NN --> ( J X. NN0 ) )
6	3, 4, 5	mp2b 8	. . . . . 6 |- `' F : NN --> ( J X. NN0 )
7	6	ffvelcdmi 5696	. . . . 5 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6224	. . . . 5 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( 2nd ` ( `' F ` A ) ) e. NN0 )
9	7, 8	syl 14	. . . 4 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. NN0 )
10	9	nn0zd 9446	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9152	. . . . 5 |- 2 e. NN
12	11	a1i 9	. . . 4 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9447	. . 3 |- ( A e. NN -> 2 e. ZZ )
14		dvdsmul2 11979	. . 3 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
15	10, 13, 14	syl2anc 411	. 2 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
16		xp1st 6223	. . . . . . . . . 10 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( 1st ` ( `' F ` A ) ) e. J )
17	7, 16	syl 14	. . . . . . . . 9 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. J )
18		breq2 4037	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
19	18	notbid 668	. . . . . . . . . . 11 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
20	19, 1	elrab2 2923	. . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
21	20	simplbi 274	. . . . . . . . 9 |- ( ( 1st ` ( `' F ` A ) ) e. J -> ( 1st ` ( `' F ` A ) ) e. NN )
22	17, 21	syl 14	. . . . . . . 8 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. NN )
23	22	nnsqcld 10786	. . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
24	20	simprbi 275	. . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J -> -. 2 || ( 1st ` ( `' F ` A ) ) )
25	17, 24	syl 14	. . . . . . . . 9 |- ( A e. NN -> -. 2 || ( 1st ` ( `' F ` A ) ) )
26		2prm 12295	. . . . . . . . . 10 |- 2 e. Prime
27	22	nnzd 9447	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
28		euclemma 12314	. . . . . . . . . . 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 ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) ) )
29		oridm 758	. . . . . . . . . . 11 |- ( ( 2 || ( 1st ` ( `' F ` A ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) )
30	28, 29	bitrdi 196	. . . . . . . . . 10 |- ( ( 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 ) ) ) )
31	26, 27, 27, 30	mp3an2i 1353	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
32	25, 31	mtbird 674	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
33	22	nncnd 9004	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
34	33	sqvald 10762	. . . . . . . . 9 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
35	34	breq2d 4045	. . . . . . . 8 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
36	32, 35	mtbird 674	. . . . . . 7 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
37		breq2 4037	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
38	37	notbid 668	. . . . . . . 8 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
39	38, 1	elrab2 2923	. . . . . . 7 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
40	23, 36, 39	sylanbrc 417	. . . . . 6 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
41	12	nnnn0d 9302	. . . . . . 7 |- ( A e. NN -> 2 e. NN0 )
42	9, 41	nn0mulcld 9307	. . . . . 6 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
43		opelxp 4693	. . . . . 6 |- ( <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J /\ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 ) )
44	40, 42, 43	sylanbrc 417	. . . . 5 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) )
45	12	nncnd 9004	. . . . . . . . 9 |- ( A e. NN -> 2 e. CC )
46	45, 41, 9	expmuld 10768	. . . . . . . 8 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
47	46	oveq1d 5937	. . . . . . 7 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
48	12, 42	nnexpcld 10787	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. NN )
49	48, 23	nnmulcld 9039	. . . . . . . 8 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
50		oveq2 5930	. . . . . . . . 9 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
51		oveq2 5930	. . . . . . . . . 10 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( 2 ^ y ) = ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
52	51	oveq1d 5937	. . . . . . . . 9 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
53	50, 52, 2	ovmpog 6057	. . . . . . . 8 |- ( ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J /\ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 /\ ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN ) -> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
54	40, 42, 49, 53	syl3anc 1249	. . . . . . 7 |- ( A e. NN -> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
55		f1ocnvfv2 5825	. . . . . . . . . . . . 13 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
56	3, 55	mpan 424	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
57		1st2nd2 6233	. . . . . . . . . . . . . 14 |- ( ( `' F ` A ) e. ( J X. NN0 ) -> ( `' F ` A ) = <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
58	7, 57	syl 14	. . . . . . . . . . . . 13 |- ( A e. NN -> ( `' F ` A ) = <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
59	58	fveq2d 5562	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
60	56, 59	eqtr3d 2231	. . . . . . . . . . 11 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
61		df-ov 5925	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
62	60, 61	eqtr4di 2247	. . . . . . . . . 10 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
63	12, 9	nnexpcld 10787	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
64	63, 22	nnmulcld 9039	. . . . . . . . . . 11 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
65		oveq2 5930	. . . . . . . . . . . 12 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
66		oveq2 5930	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
67	66	oveq1d 5937	. . . . . . . . . . . 12 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
68	65, 67, 2	ovmpog 6057	. . . . . . . . . . 11 |- ( ( ( 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 ) ) ) )
69	17, 9, 64, 68	syl3anc 1249	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
70	62, 69	eqtrd 2229	. . . . . . . . 9 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
71	70	oveq1d 5937	. . . . . . . 8 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
72	63	nncnd 9004	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
73	72, 33	sqmuld 10777	. . . . . . . 8 |- ( A e. NN -> ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
74	71, 73	eqtrd 2229	. . . . . . 7 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
75	47, 54, 74	3eqtr4rd 2240	. . . . . 6 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
76		df-ov 5925	. . . . . 6 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. )
77	75, 76	eqtr2di 2246	. . . . 5 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) )
78		f1ocnvfv 5826	. . . . . 6 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) ) -> ( ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) -> ( `' F ` ( A ^ 2 ) ) = <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
79	3, 78	mpan 424	. . . . 5 |- ( <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) -> ( ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) -> ( `' F ` ( A ^ 2 ) ) = <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
80	44, 77, 79	sylc 62	. . . 4 |- ( A e. NN -> ( `' F ` ( A ^ 2 ) ) = <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. )
81	80	fveq2d 5562	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
82		op2ndg 6209	. . . 4 |- ( ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J /\ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 ) -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
83	40, 42, 82	syl2anc 411	. . 3 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
84	81, 83	eqtrd 2229	. 2 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
85	15, 84	breqtrrd 4061	1 |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   {crab 2479   <.cop 3625   class class class wbr 4033   X. cxp 4661   `'ccnv 4662   -->wf 5254   -1-1-onto->wf1o 5257   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197   x. cmul 7884   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ^cexp 10630   || cdvds 11952   Primecprime 12275

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

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-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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-sup 7050  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276

This theorem is referenced by:  sqne2sq  12345