Theorem sqpweven 12175

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 12174	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5475	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5462	. . . . . . 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 5651	. . . . 5 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6167	. . . . 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 9373	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9080	. . . . 5 |- 2 e. NN
12	11	a1i 9	. . . 4 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9374	. . 3 |- ( A e. NN -> 2 e. ZZ )
14		dvdsmul2 11821	. . 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 6166	. . . . . . . . . 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 4008	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
19	18	notbid 667	. . . . . . . . . . 11 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
20	19, 1	elrab2 2897	. . . . . . . . . 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 10675	. . . . . . 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 12127	. . . . . . . . . 10 |- 2 e. Prime
27	22	nnzd 9374	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
28		euclemma 12146	. . . . . . . . . . 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 757	. . . . . . . . . . 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 1342	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
32	25, 31	mtbird 673	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
33	22	nncnd 8933	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
34	33	sqvald 10651	. . . . . . . . 9 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
35	34	breq2d 4016	. . . . . . . 8 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
36	32, 35	mtbird 673	. . . . . . 7 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
37		breq2 4008	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
38	37	notbid 667	. . . . . . . 8 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
39	38, 1	elrab2 2897	. . . . . . 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 9229	. . . . . . 7 |- ( A e. NN -> 2 e. NN0 )
42	9, 41	nn0mulcld 9234	. . . . . 6 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
43		opelxp 4657	. . . . . 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 8933	. . . . . . . . 9 |- ( A e. NN -> 2 e. CC )
46	45, 41, 9	expmuld 10657	. . . . . . . 8 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
47	46	oveq1d 5890	. . . . . . 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 10676	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. NN )
49	48, 23	nnmulcld 8968	. . . . . . . 8 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
50		oveq2 5883	. . . . . . . . 9 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
51		oveq2 5883	. . . . . . . . . 10 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( 2 ^ y ) = ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
52	51	oveq1d 5890	. . . . . . . . 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 6009	. . . . . . . 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 1238	. . . . . . 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 5779	. . . . . . . . . . . . 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 6176	. . . . . . . . . . . . . 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 5520	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
60	56, 59	eqtr3d 2212	. . . . . . . . . . 11 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
61		df-ov 5878	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
62	60, 61	eqtr4di 2228	. . . . . . . . . 10 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
63	12, 9	nnexpcld 10676	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
64	63, 22	nnmulcld 8968	. . . . . . . . . . 11 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
65		oveq2 5883	. . . . . . . . . . . 12 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
66		oveq2 5883	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
67	66	oveq1d 5890	. . . . . . . . . . . 12 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
68	65, 67, 2	ovmpog 6009	. . . . . . . . . . 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 1238	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
70	62, 69	eqtrd 2210	. . . . . . . . 9 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
71	70	oveq1d 5890	. . . . . . . 8 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
72	63	nncnd 8933	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
73	72, 33	sqmuld 10666	. . . . . . . 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 2210	. . . . . . 7 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
75	47, 54, 74	3eqtr4rd 2221	. . . . . 6 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
76		df-ov 5878	. . . . . 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 2227	. . . . 5 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) )
78		f1ocnvfv 5780	. . . . . 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 5520	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
82		op2ndg 6152	. . . 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 2210	. 2 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
85	15, 84	breqtrrd 4032	1 |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )

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   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-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