Theorem sqpweven 12953

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 12952	. . . . . . 7 |- F : ( J X. NN0 ) -1-1-onto-> NN
4		f1ocnv 5652	. . . . . . 7 |- ( F : ( J X. NN0 ) -1-1-onto-> NN -> `' F : NN -1-1-onto-> ( J X. NN0 ) )
5		f1of 5639	. . . . . . 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 5842	. . . . 5 |- ( A e. NN -> ( `' F ` A ) e. ( J X. NN0 ) )
8		xp2nd 6400	. . . . 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 9766	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` A ) ) e. ZZ )
11		2nn 9466	. . . . 5 |- 2 e. NN
12	11	a1i 9	. . . 4 |- ( A e. NN -> 2 e. NN )
13	12	nnzd 9767	. . 3 |- ( A e. NN -> 2 e. ZZ )
14		dvdsmul2 12581	. . 3 |- ( ( ( 2nd ` ( `' F ` A ) ) e. ZZ /\ 2 e. ZZ ) -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
15	10, 13, 14	syl2anc 415	. 2 |- ( A e. NN -> 2 || ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
16		xp1st 6399	. . . . . . . . . 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 4134	. . . . . . . . . . . 12 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( 2 || z <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
19	18	notbid 677	. . . . . . . . . . 11 |- ( z = ( 1st ` ( `' F ` A ) ) -> ( -. 2 || z <-> -. 2 || ( 1st ` ( `' F ` A ) ) ) )
20	19, 1	elrab2 2985	. . . . . . . . . 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 11132	. . . . . . 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 12905	. . . . . . . . . 10 |- 2 e. Prime
27	22	nnzd 9767	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. ZZ )
28		euclemma 12924	. . . . . . . . . . 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 769	. . . . . . . . . . 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 1383	. . . . . . . . 9 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) <-> 2 || ( 1st ` ( `' F ` A ) ) ) )
32	25, 31	mtbird 684	. . . . . . . 8 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
33	22	nncnd 9318	. . . . . . . . . 10 |- ( A e. NN -> ( 1st ` ( `' F ` A ) ) e. CC )
34	33	sqvald 11108	. . . . . . . . 9 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) = ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) )
35	34	breq2d 4142	. . . . . . . 8 |- ( A e. NN -> ( 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) <-> 2 || ( ( 1st ` ( `' F ` A ) ) x. ( 1st ` ( `' F ` A ) ) ) ) )
36	32, 35	mtbird 684	. . . . . . 7 |- ( A e. NN -> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) )
37		breq2 4134	. . . . . . . . 9 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( 2 || z <-> 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
38	37	notbid 677	. . . . . . . 8 |- ( z = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( -. 2 || z <-> -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
39	38, 1	elrab2 2985	. . . . . . 7 |- ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J <-> ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. NN /\ -. 2 || ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
40	23, 36, 39	sylanbrc 421	. . . . . 6 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) ^ 2 ) e. J )
41	12	nnnn0d 9620	. . . . . . 7 |- ( A e. NN -> 2 e. NN0 )
42	9, 41	nn0mulcld 9625	. . . . . 6 |- ( A e. NN -> ( ( 2nd ` ( `' F ` A ) ) x. 2 ) e. NN0 )
43		opelxp 4804	. . . . . 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 421	. . . . 5 |- ( A e. NN -> <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. e. ( J X. NN0 ) )
45	12	nncnd 9318	. . . . . . . . 9 |- ( A e. NN -> 2 e. CC )
46	45, 41, 9	expmuld 11114	. . . . . . . 8 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) )
47	46	oveq1d 6100	. . . . . . 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 11133	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) e. NN )
49	48, 23	nnmulcld 9353	. . . . . . . 8 |- ( A e. NN -> ( ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) e. NN )
50		oveq2 6093	. . . . . . . . 9 |- ( x = ( ( 1st ` ( `' F ` A ) ) ^ 2 ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
51		oveq2 6093	. . . . . . . . . 10 |- ( y = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) -> ( 2 ^ y ) = ( 2 ^ ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
52	51	oveq1d 6100	. . . . . . . . 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 6223	. . . . . . . 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 1278	. . . . . . 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 5984	. . . . . . . . . . . . 13 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ A e. NN ) -> ( F ` ( `' F ` A ) ) = A )
56	3, 55	mpan 428	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = A )
57		1st2nd2 6409	. . . . . . . . . . . . . 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 5699	. . . . . . . . . . . 12 |- ( A e. NN -> ( F ` ( `' F ` A ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
60	56, 59	eqtr3d 2273	. . . . . . . . . . 11 |- ( A e. NN -> A = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. ) )
61		df-ov 6088	. . . . . . . . . . 11 |- ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( F ` <. ( 1st ` ( `' F ` A ) ) , ( 2nd ` ( `' F ` A ) ) >. )
62	60, 61	eqtr4di 2289	. . . . . . . . . 10 |- ( A e. NN -> A = ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) )
63	12, 9	nnexpcld 11133	. . . . . . . . . . . 12 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. NN )
64	63, 22	nnmulcld 9353	. . . . . . . . . . 11 |- ( A e. NN -> ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) e. NN )
65		oveq2 6093	. . . . . . . . . . . 12 |- ( x = ( 1st ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. x ) = ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) )
66		oveq2 6093	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( 2 ^ y ) = ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) )
67	66	oveq1d 6100	. . . . . . . . . . . 12 |- ( y = ( 2nd ` ( `' F ` A ) ) -> ( ( 2 ^ y ) x. ( 1st ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
68	65, 67, 2	ovmpog 6223	. . . . . . . . . . 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 1278	. . . . . . . . . 10 |- ( A e. NN -> ( ( 1st ` ( `' F ` A ) ) F ( 2nd ` ( `' F ` A ) ) ) = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
70	62, 69	eqtrd 2271	. . . . . . . . 9 |- ( A e. NN -> A = ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) )
71	70	oveq1d 6100	. . . . . . . 8 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) x. ( 1st ` ( `' F ` A ) ) ) ^ 2 ) )
72	63	nncnd 9318	. . . . . . . . 9 |- ( A e. NN -> ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) e. CC )
73	72, 33	sqmuld 11123	. . . . . . . 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 2271	. . . . . . 7 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 2 ^ ( 2nd ` ( `' F ` A ) ) ) ^ 2 ) x. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) ) )
75	47, 54, 74	3eqtr4rd 2282	. . . . . 6 |- ( A e. NN -> ( A ^ 2 ) = ( ( ( 1st ` ( `' F ` A ) ) ^ 2 ) F ( ( 2nd ` ( `' F ` A ) ) x. 2 ) ) )
76		df-ov 6088	. . . . . 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 2288	. . . . 5 |- ( A e. NN -> ( F ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( A ^ 2 ) )
78		f1ocnvfv 5985	. . . . . 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 428	. . . . 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 5699	. . 3 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) )
82		op2ndg 6385	. . . 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 415	. . 3 |- ( A e. NN -> ( 2nd ` <. ( ( 1st ` ( `' F ` A ) ) ^ 2 ) , ( ( 2nd ` ( `' F ` A ) ) x. 2 ) >. ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
84	81, 83	eqtrd 2271	. 2 |- ( A e. NN -> ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) = ( ( 2nd ` ( `' F ` A ) ) x. 2 ) )
85	15, 84	breqtrrd 4158	1 |- ( A e. NN -> 2 || ( 2nd ` ( `' F ` ( A ^ 2 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   <.cop 3712   class class class wbr 4130   X. cxp 4772   `'ccnv 4773   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   x. cmul 8184   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ^cexp 10975   || cdvds 12554   Primecprime 12885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886

This theorem is used by:  sqne2sq  12955