Theorem perfect 15517

Description: The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer N is a perfect number (that is, its divisor sum is 2 N) if and only if it is of the form 2 ^ ( p - 1 ) x. ( 2 ^ p - 1 ), where 2 ^ p - 1 is prime (a Mersenne prime), and therefore p is also prime, see mersenne 15513. This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)

Assertion

Ref	Expression
perfect	|- ( ( N e. NN /\ 2 || N ) -> ( ( 1 sigma N ) = ( 2 x. N ) <-> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) )

Distinct variable group:   N, p

Proof of Theorem perfect

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> 2 || N )
2		2prm 12493	. . . . . . . 8 |- 2 e. Prime
3		simpll 527	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. NN )
4		pcelnn 12688	. . . . . . . 8 |- ( ( 2 e. Prime /\ N e. NN ) -> ( ( 2 pCnt N ) e. NN <-> 2 || N ) )
5	2, 3, 4	sylancr 414	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) e. NN <-> 2 || N ) )
6	1, 5	mpbird 167	. . . . . 6 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN )
7	6	nnzd 9501	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. ZZ )
8	7	peano2zd 9505	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) + 1 ) e. ZZ )
9		pcdvds 12682	. . . . . . . . 9 |- ( ( 2 e. Prime /\ N e. NN ) -> ( 2 ^ ( 2 pCnt N ) ) || N )
10	2, 3, 9	sylancr 414	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) || N )
11		2nn 9205	. . . . . . . . . 10 |- 2 e. NN
12	6	nnnn0d 9355	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN0 )
13		nnexpcl 10704	. . . . . . . . . 10 |- ( ( 2 e. NN /\ ( 2 pCnt N ) e. NN0 ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN )
14	11, 12, 13	sylancr 414	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN )
15		nndivdvds 12151	. . . . . . . . 9 |- ( ( N e. NN /\ ( 2 ^ ( 2 pCnt N ) ) e. NN ) -> ( ( 2 ^ ( 2 pCnt N ) ) || N <-> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN ) )
16	3, 14, 15	syl2anc 411	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( 2 pCnt N ) ) || N <-> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN ) )
17	10, 16	mpbid 147	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN )
18		pcndvds2 12686	. . . . . . . 8 |- ( ( 2 e. Prime /\ N e. NN ) -> -. 2 || ( N / ( 2 ^ ( 2 pCnt N ) ) ) )
19	2, 3, 18	sylancr 414	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> -. 2 || ( N / ( 2 ^ ( 2 pCnt N ) ) ) )
20		simpr 110	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 1 sigma N ) = ( 2 x. N ) )
21		nncn 9051	. . . . . . . . . . 11 |- ( N e. NN -> N e. CC )
22	21	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. CC )
23	14	nncnd 9057	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. CC )
24	14	nnap0d 9089	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) # 0 )
25	22, 23, 24	divcanap2d 8872	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) = N )
26	25	oveq2d 5967	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 1 sigma ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) ) = ( 1 sigma N ) )
27	25	oveq2d 5967	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 x. ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) ) = ( 2 x. N ) )
28	20, 26, 27	3eqtr4d 2249	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 1 sigma ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) ) = ( 2 x. ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) ) )
29	6, 17, 19, 28	perfectlem2 15516	. . . . . 6 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. Prime /\ ( N / ( 2 ^ ( 2 pCnt N ) ) ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) )
30	29	simprd 114	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) )
31	29	simpld 112	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. Prime )
32	30, 31	eqeltrrd 2284	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime )
33	6	nncnd 9057	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. CC )
34		ax-1cn 8025	. . . . . . . . 9 |- 1 e. CC
35		pncan 8285	. . . . . . . . 9 |- ( ( ( 2 pCnt N ) e. CC /\ 1 e. CC ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) )
36	33, 34, 35	sylancl 413	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) )
37	36	eqcomd 2212	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
38	37	oveq2d 5967	. . . . . 6 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) )
39	38, 30	oveq12d 5969	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) )
40	25, 39	eqtr3d 2241	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) )
41		oveq2 5959	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ p ) = ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) )
42	41	oveq1d 5966	. . . . . . 7 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ p ) - 1 ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) )
43	42	eleq1d 2275	. . . . . 6 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( 2 ^ p ) - 1 ) e. Prime <-> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) )
44		oveq1 5958	. . . . . . . . 9 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( p - 1 ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
45	44	oveq2d 5967	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ ( p - 1 ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) )
46	45, 42	oveq12d 5969	. . . . . . 7 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) )
47	46	eqeq2d 2218	. . . . . 6 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) <-> N = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) ) )
48	43, 47	anbi12d 473	. . . . 5 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) <-> ( ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) ) ) )
49	48	rspcev 2878	. . . 4 |- ( ( ( ( 2 pCnt N ) + 1 ) e. ZZ /\ ( ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) x. ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) ) ) -> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
50	8, 32, 40, 49	syl12anc 1248	. . 3 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
51	50	ex 115	. 2 |- ( ( N e. NN /\ 2 || N ) -> ( ( 1 sigma N ) = ( 2 x. N ) -> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) )
52		perfect1 15514	. . . . . 6 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) = ( ( 2 ^ p ) x. ( ( 2 ^ p ) - 1 ) ) )
53		2cn 9114	. . . . . . . . 9 |- 2 e. CC
54		mersenne 15513	. . . . . . . . . 10 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. Prime )
55		prmnn 12476	. . . . . . . . . 10 |- ( p e. Prime -> p e. NN )
56	54, 55	syl 14	. . . . . . . . 9 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. NN )
57		expm1t 10719	. . . . . . . . 9 |- ( ( 2 e. CC /\ p e. NN ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) )
58	53, 56, 57	sylancr 414	. . . . . . . 8 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) )
59		nnm1nn0 9343	. . . . . . . . . . 11 |- ( p e. NN -> ( p - 1 ) e. NN0 )
60	56, 59	syl 14	. . . . . . . . . 10 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( p - 1 ) e. NN0 )
61		expcl 10709	. . . . . . . . . 10 |- ( ( 2 e. CC /\ ( p - 1 ) e. NN0 ) -> ( 2 ^ ( p - 1 ) ) e. CC )
62	53, 60, 61	sylancr 414	. . . . . . . . 9 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ ( p - 1 ) ) e. CC )
63		mulcom 8061	. . . . . . . . 9 |- ( ( ( 2 ^ ( p - 1 ) ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
64	62, 53, 63	sylancl 413	. . . . . . . 8 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
65	58, 64	eqtrd 2239	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
66	65	oveq1d 5966	. . . . . 6 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) x. ( ( 2 ^ p ) - 1 ) ) = ( ( 2 x. ( 2 ^ ( p - 1 ) ) ) x. ( ( 2 ^ p ) - 1 ) ) )
67		2cnd 9116	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> 2 e. CC )
68		prmnn 12476	. . . . . . . . 9 |- ( ( ( 2 ^ p ) - 1 ) e. Prime -> ( ( 2 ^ p ) - 1 ) e. NN )
69	68	adantl 277	. . . . . . . 8 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. NN )
70	69	nncnd 9057	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. CC )
71	67, 62, 70	mulassd 8103	. . . . . 6 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 x. ( 2 ^ ( p - 1 ) ) ) x. ( ( 2 ^ p ) - 1 ) ) = ( 2 x. ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
72	52, 66, 71	3eqtrd 2243	. . . . 5 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) = ( 2 x. ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
73		oveq2 5959	. . . . . 6 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
74		oveq2 5959	. . . . . 6 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 2 x. N ) = ( 2 x. ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
75	73, 74	eqeq12d 2221	. . . . 5 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( ( 1 sigma N ) = ( 2 x. N ) <-> ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) = ( 2 x. ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) )
76	72, 75	syl5ibrcom 157	. . . 4 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 2 x. N ) ) )
77	76	impr 379	. . 3 |- ( ( p e. ZZ /\ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) -> ( 1 sigma N ) = ( 2 x. N ) )
78	77	rexlimiva 2619	. 2 |- ( E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) -> ( 1 sigma N ) = ( 2 x. N ) )
79	51, 78	impbid1 142	1 |- ( ( N e. NN /\ 2 || N ) -> ( ( 1 sigma N ) = ( 2 x. N ) <-> E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   E.wrex 2486   class class class wbr 4047  (class class class)co 5951   CCcc 7930   1c1 7933   + caddc 7935   x. cmul 7937   - cmin 8250   / cdiv 8752   NNcn 9043   2c2 9094   NN0cn0 9302   ZZcz 9379   ^cexp 10690   || cdvds 12142   Primecprime 12473   pCnt cpc 12651   sigma csgm 15497

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052  ax-pre-suploc 8053  ax-addf 8054  ax-mulf 8055

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-tp 3642  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-disj 4024  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-isom 5285  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-of 6165  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-frec 6484  df-1o 6509  df-2o 6510  df-oadd 6513  df-er 6627  df-map 6744  df-pm 6745  df-en 6835  df-dom 6836  df-fin 6837  df-sup 7093  df-inf 7094  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-xnn0 9366  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-xneg 9901  df-xadd 9902  df-ioo 10021  df-ico 10023  df-icc 10024  df-fz 10138  df-fzo 10272  df-fl 10420  df-mod 10475  df-seqfrec 10600  df-exp 10691  df-fac 10878  df-bc 10900  df-ihash 10928  df-shft 11170  df-cj 11197  df-re 11198  df-im 11199  df-rsqrt 11353  df-abs 11354  df-clim 11634  df-sumdc 11709  df-ef 12003  df-e 12004  df-dvds 12143  df-gcd 12319  df-prm 12474  df-pc 12652  df-rest 13117  df-topgen 13136  df-psmet 14349  df-xmet 14350  df-met 14351  df-bl 14352  df-mopn 14353  df-top 14514  df-topon 14527  df-bases 14559  df-ntr 14612  df-cn 14704  df-cnp 14705  df-tx 14769  df-cncf 15087  df-limced 15172  df-dvap 15173  df-relog 15374  df-rpcxp 15375  df-sgm 15498

This theorem is referenced by: (None)