Theorem perfect 16029

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 16025. 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 533	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> 2 || N )
2		2prm 12883	. . . . . . . 8 |- 2 e. Prime
3		simpll 531	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. NN )
4		pcelnn 13078	. . . . . . . 8 |- ( ( 2 e. Prime /\ N e. NN ) -> ( ( 2 pCnt N ) e. NN <-> 2 || N ) )
5	2, 3, 4	sylancr 418	. . . . . . 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 9746	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. ZZ )
8	7	peano2zd 9750	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) + 1 ) e. ZZ )
9		pcdvds 13072	. . . . . . . . 9 |- ( ( 2 e. Prime /\ N e. NN ) -> ( 2 ^ ( 2 pCnt N ) ) || N )
10	2, 3, 9	sylancr 418	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) || N )
11		2nn 9445	. . . . . . . . . 10 |- 2 e. NN
12	6	nnnn0d 9599	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN0 )
13		nnexpcl 10967	. . . . . . . . . 10 |- ( ( 2 e. NN /\ ( 2 pCnt N ) e. NN0 ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN )
14	11, 12, 13	sylancr 418	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN )
15		nndivdvds 12541	. . . . . . . . 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 415	. . . . . . . 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 13076	. . . . . . . 8 |- ( ( 2 e. Prime /\ N e. NN ) -> -. 2 || ( N / ( 2 ^ ( 2 pCnt N ) ) ) )
19	2, 3, 18	sylancr 418	. . . . . . 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 9291	. . . . . . . . . . 11 |- ( N e. NN -> N e. CC )
22	21	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. CC )
23	14	nncnd 9297	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. CC )
24	14	nnap0d 9329	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) # 0 )
25	22, 23, 24	divcanap2d 9112	. . . . . . . . 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 6091	. . . . . . . 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 6091	. . . . . . . 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 2281	. . . . . . 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 16028	. . . . . 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 2316	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime )
33	6	nncnd 9297	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. CC )
34		ax-1cn 8262	. . . . . . . . 9 |- 1 e. CC
35		pncan 8522	. . . . . . . . 9 |- ( ( ( 2 pCnt N ) e. CC /\ 1 e. CC ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) )
36	33, 34, 35	sylancl 417	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) )
37	36	eqcomd 2244	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
38	37	oveq2d 6091	. . . . . 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 6093	. . . . 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 2273	. . . 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 6083	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ p ) = ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) )
42	41	oveq1d 6090	. . . . . . 7 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ p ) - 1 ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) )
43	42	eleq1d 2307	. . . . . 6 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( 2 ^ p ) - 1 ) e. Prime <-> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) )
44		oveq1 6082	. . . . . . . . 9 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( p - 1 ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
45	44	oveq2d 6091	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ ( p - 1 ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) )
46	45, 42	oveq12d 6093	. . . . . . 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 2250	. . . . . 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 477	. . . . 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 2929	. . . 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 1276	. . 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 16026	. . . . . 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 9354	. . . . . . . . 9 |- 2 e. CC
54		mersenne 16025	. . . . . . . . . 10 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. Prime )
55		prmnn 12866	. . . . . . . . . 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 10982	. . . . . . . . 9 |- ( ( 2 e. CC /\ p e. NN ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) )
58	53, 56, 57	sylancr 418	. . . . . . . 8 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) )
59		nnm1nn0 9583	. . . . . . . . . . 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 10972	. . . . . . . . . 10 |- ( ( 2 e. CC /\ ( p - 1 ) e. NN0 ) -> ( 2 ^ ( p - 1 ) ) e. CC )
62	53, 60, 61	sylancr 418	. . . . . . . . 9 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ ( p - 1 ) ) e. CC )
63		mulcom 8298	. . . . . . . . 9 |- ( ( ( 2 ^ ( p - 1 ) ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
64	62, 53, 63	sylancl 417	. . . . . . . 8 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
65	58, 64	eqtrd 2271	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
66	65	oveq1d 6090	. . . . . 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 9356	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> 2 e. CC )
68		prmnn 12866	. . . . . . . . 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 9297	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. CC )
71	67, 62, 70	mulassd 8339	. . . . . 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 2275	. . . . 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 6083	. . . . . 6 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
74		oveq2 6083	. . . . . 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 2253	. . . . 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 2663	. 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 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   1c1 8170   + caddc 8172   x. cmul 8174   - cmin 8487   / cdiv 8992   NNcn 9283   2c2 9334   NN0cn0 9542   ZZcz 9623   ^cexp 10953   || cdvds 12532   Primecprime 12863   pCnt cpc 13041   sigma csgm 16009

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-pre-suploc 8290  ax-addf 8291  ax-mulf 8292

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-2o 6678  df-oadd 6681  df-er 6797  df-map 6914  df-pm 6915  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-xnn0 9610  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-ioo 10273  df-ico 10275  df-icc 10276  df-fz 10391  df-fzo 10528  df-fl 10683  df-mod 10738  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-e 12394  df-dvds 12533  df-gcd 12709  df-prm 12864  df-pc 13042  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681  df-relog 15882  df-rpcxp 15883  df-sgm 16010

This theorem is referenced by: (None)