Theorem sqpweven 10778

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 10777	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5191	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5178	. . . . . . 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 5354	. . . . 5 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 5845	. . . . 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 8618	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 8330	. . . . 5 |- 2 e. NN
12	11	a1i 9	. . . 4 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 8619	. . 3 |- ( A e. NN -> 2 e. ZZ )
14		dvdsmul2 10444	. . 3 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
15	10, 13, 14	syl2anc 403	. 2 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
16		xp1st 5844	. . . . . . . . . 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 3809	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
19	18	notbid 625	. . . . . . . . . . 11 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
20	19, 1	elrab2 2760	. . . . . . . . . 10 |- ( ( 1st ` ( `' F ` A ) ) e. J <-> ( ( 1st ` ( `' F ` A ) ) e. NN /\ -. 2 || ( 1st ` ( `' F ` A ) ) ) )
21	20	simplbi 268	. . . . . . . . 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 9793	. . . . . . 7 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN )
24	20	simprbi 269	. . . . . . . . . 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 10734	. . . . . . . . . 10 |- 2 e. Prime
27	22	nnzd 8619	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
28		euclemma 10750	. . . . . . . . . . 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 707	. . . . . . . . . . 11 |- ( ( 2 || ( 1st ` ( `' F ` A ) ) \/ 2 || ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) )
30	28, 29	syl6bb 194	. . . . . . . . . 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 1274	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
32	25, 31	mtbird 631	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
33	22	nncnd 8190	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
34	33	sqvald 9769	. . . . . . . . 9 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
35	34	breq2d 3817	. . . . . . . 8 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
36	32, 35	mtbird 631	. . . . . . 7 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
37		breq2 3809	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
38	37	notbid 625	. . . . . . . 8 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
39	38, 1	elrab2 2760	. . . . . . 7 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
40	23, 36, 39	sylanbrc 408	. . . . . 6 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
41	12	nnnn0d 8478	. . . . . . 7 |- ( A e. NN -> 2 e. NN0 )
42	9, 41	nn0mulcld 8483	. . . . . 6 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
43		opelxp 4420	. . . . . 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 408	. . . . 5 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) )
45	12	nncnd 8190	. . . . . . . . 9 |- ( A e. NN -> 2 e. CC )
46	45, 41, 9	expmuld 9775	. . . . . . . 8 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
47	46	oveq1d 5579	. . . . . . 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 9794	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. NN )
49	48, 23	nnmulcld 8224	. . . . . . . 8 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
50		oveq2 5572	. . . . . . . . 9 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
51		oveq2 5572	. . . . . . . . . 10 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( 2 ^ y ) = ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
52	51	oveq1d 5579	. . . . . . . . 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	ovmpt2g 5687	. . . . . . . 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 1170	. . . . . . 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 5470	. . . . . . . . . . . . 13 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
56	3, 55	mpan 415	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
57		1st2nd2 5853	. . . . . . . . . . . . . 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 5234	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
60	56, 59	eqtr3d 2117	. . . . . . . . . . 11 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
61		df-ov 5567	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
62	60, 61	syl6eqr 2133	. . . . . . . . . 10 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
63	12, 9	nnexpcld 9794	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
64	63, 22	nnmulcld 8224	. . . . . . . . . . 11 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
65		oveq2 5572	. . . . . . . . . . . 12 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
66		oveq2 5572	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
67	66	oveq1d 5579	. . . . . . . . . . . 12 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
68	65, 67, 2	ovmpt2g 5687	. . . . . . . . . . 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 1170	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
70	62, 69	eqtrd 2115	. . . . . . . . 9 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
71	70	oveq1d 5579	. . . . . . . 8 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
72	63	nncnd 8190	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
73	72, 33	sqmuld 9784	. . . . . . . 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 2115	. . . . . . 7 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
75	47, 54, 74	3eqtr4rd 2126	. . . . . 6 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
76		df-ov 5567	. . . . . 6 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. )
77	75, 76	syl6req 2132	. . . . 5 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) )
78		f1ocnvfv 5471	. . . . . 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 415	. . . . 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 61	. . . 4 |- ( A e. NN -> ( `' F ` ( A ^ 2 ) ) = <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. )
81	80	fveq2d 5234	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
82		op2ndg 5830	. . . 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 403	. . 3 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
84	81, 83	eqtrd 2115	. 2 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
85	15, 84	breqtrrd 3831	1 |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 662   /\ w3a 920   = wceq 1285   e. wcel 1434   {crab 2357   <.cop 3419   class class class wbr 3805   X. cxp 4389   `'ccnv 4390   -->wf 4948   -1-1-onto->wf1o 4951   ` cfv 4952  (class class class)co 5564   |-> cmpt2 5566   1stc1st 5817   2ndc2nd 5818   x. cmul 7118   NNcn 8176   2c2 8226   NN0cn0 8425   ZZcz 8502   ^cexp 9642   || cdvds 10421   Primecprime 10714

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226  ax-arch 7227  ax-caucvg 7228

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-1o 6086  df-2o 6087  df-er 6194  df-en 6310  df-sup 6492  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-3 8236  df-4 8237  df-n0 8426  df-z 8503  df-uz 8771  df-q 8856  df-rp 8886  df-fz 9176  df-fzo 9300  df-fl 9422  df-mod 9475  df-iseq 9592  df-iexp 9643  df-cj 9948  df-re 9949  df-im 9950  df-rsqrt 10103  df-abs 10104  df-dvds 10422  df-gcd 10564  df-prm 10715

This theorem is referenced by:  sqne2sq  10780