Theorem sqpweven 12129

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 12128	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5455	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5442	. . . . . . 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	ffvelrni 5630	. . . . 5 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6145	. . . . 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 9332	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9039	. . . . 5 |- 2 e. NN
12	11	a1i 9	. . . 4 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9333	. . 3 |- ( A e. NN -> 2 e. ZZ )
14		dvdsmul2 11776	. . 3 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
15	10, 13, 14	syl2anc 409	. 2 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
16		xp1st 6144	. . . . . . . . . 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 3993	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
19	18	notbid 662	. . . . . . . . . . 11 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
20	19, 1	elrab2 2889	. . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
21	20	simplbi 272	. . . . . . . . 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 10630	. . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
24	20	simprbi 273	. . . . . . . . . 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 12081	. . . . . . . . . 10 |- 2 e. Prime
27	22	nnzd 9333	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
28		euclemma 12100	. . . . . . . . . . 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 752	. . . . . . . . . . 11 |- ( ( 2 || ( 1st ` ( `' F ` A ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) )
30	28, 29	bitrdi 195	. . . . . . . . . 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 1337	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
32	25, 31	mtbird 668	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
33	22	nncnd 8892	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
34	33	sqvald 10606	. . . . . . . . 9 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
35	34	breq2d 4001	. . . . . . . 8 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
36	32, 35	mtbird 668	. . . . . . 7 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
37		breq2 3993	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
38	37	notbid 662	. . . . . . . 8 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
39	38, 1	elrab2 2889	. . . . . . 7 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
40	23, 36, 39	sylanbrc 415	. . . . . 6 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
41	12	nnnn0d 9188	. . . . . . 7 |- ( A e. NN -> 2 e. NN0 )
42	9, 41	nn0mulcld 9193	. . . . . 6 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
43		opelxp 4641	. . . . . 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 415	. . . . 5 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) )
45	12	nncnd 8892	. . . . . . . . 9 |- ( A e. NN -> 2 e. CC )
46	45, 41, 9	expmuld 10612	. . . . . . . 8 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
47	46	oveq1d 5868	. . . . . . 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 10631	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. NN )
49	48, 23	nnmulcld 8927	. . . . . . . 8 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
50		oveq2 5861	. . . . . . . . 9 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
51		oveq2 5861	. . . . . . . . . 10 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( 2 ^ y ) = ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
52	51	oveq1d 5868	. . . . . . . . 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 5987	. . . . . . . 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 1233	. . . . . . 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 5757	. . . . . . . . . . . . 13 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
56	3, 55	mpan 422	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
57		1st2nd2 6154	. . . . . . . . . . . . . 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 5500	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
60	56, 59	eqtr3d 2205	. . . . . . . . . . 11 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
61		df-ov 5856	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
62	60, 61	eqtr4di 2221	. . . . . . . . . 10 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
63	12, 9	nnexpcld 10631	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
64	63, 22	nnmulcld 8927	. . . . . . . . . . 11 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
65		oveq2 5861	. . . . . . . . . . . 12 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
66		oveq2 5861	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
67	66	oveq1d 5868	. . . . . . . . . . . 12 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
68	65, 67, 2	ovmpog 5987	. . . . . . . . . . 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 1233	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
70	62, 69	eqtrd 2203	. . . . . . . . 9 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
71	70	oveq1d 5868	. . . . . . . 8 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
72	63	nncnd 8892	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
73	72, 33	sqmuld 10621	. . . . . . . 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 2203	. . . . . . 7 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
75	47, 54, 74	3eqtr4rd 2214	. . . . . 6 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
76		df-ov 5856	. . . . . 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 2220	. . . . 5 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) )
78		f1ocnvfv 5758	. . . . . 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 422	. . . . 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 5500	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
82		op2ndg 6130	. . . 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 409	. . 3 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
84	81, 83	eqtrd 2203	. 2 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
85	15, 84	breqtrrd 4017	1 |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 703   /\ w3a 973   = wceq 1348   e. wcel 2141   {crab 2452   <.cop 3586   class class class wbr 3989   X. cxp 4609   `'ccnv 4610   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   x. cmul 7779   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ^cexp 10475   || cdvds 11749   Primecprime 12061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062

This theorem is referenced by:  sqne2sq  12131