Theorem perfect 15918

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 15914. 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 529	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> 2 || N )
2		2prm 12832	. . . . . . . 8 |- 2 e. Prime
3		simpll 527	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. NN )
4		pcelnn 13027	. . . . . . . 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 9705	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. ZZ )
8	7	peano2zd 9709	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) + 1 ) e. ZZ )
9		pcdvds 13021	. . . . . . . . 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 9404	. . . . . . . . . 10 |- 2 e. NN
12	6	nnnn0d 9558	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN0 )
13		nnexpcl 10921	. . . . . . . . . 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 12490	. . . . . . . . 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 13025	. . . . . . . 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 9250	. . . . . . . . . . 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 9256	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. CC )
24	14	nnap0d 9288	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) # 0 )
25	22, 23, 24	divcanap2d 9071	. . . . . . . . 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 6068	. . . . . . . 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 6068	. . . . . . . 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 2277	. . . . . . 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 15917	. . . . . 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 2312	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime )
33	6	nncnd 9256	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. CC )
34		ax-1cn 8225	. . . . . . . . 9 |- 1 e. CC
35		pncan 8484	. . . . . . . . 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 2240	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
38	37	oveq2d 6068	. . . . . 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 6070	. . . . 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 2269	. . . 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 6060	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ p ) = ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) )
42	41	oveq1d 6067	. . . . . . 7 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ p ) - 1 ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) )
43	42	eleq1d 2303	. . . . . 6 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( 2 ^ p ) - 1 ) e. Prime <-> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) )
44		oveq1 6059	. . . . . . . . 9 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( p - 1 ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
45	44	oveq2d 6068	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ ( p - 1 ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) )
46	45, 42	oveq12d 6070	. . . . . . 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 2246	. . . . . 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 2923	. . . 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 1272	. . 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 15915	. . . . . 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 9313	. . . . . . . . 9 |- 2 e. CC
54		mersenne 15914	. . . . . . . . . 10 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. Prime )
55		prmnn 12815	. . . . . . . . . 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 10936	. . . . . . . . 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 9542	. . . . . . . . . . 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 10926	. . . . . . . . . 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 8261	. . . . . . . . 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 2267	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
66	65	oveq1d 6067	. . . . . 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 9315	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> 2 e. CC )
68		prmnn 12815	. . . . . . . . 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 9256	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. CC )
71	67, 62, 70	mulassd 8302	. . . . . 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 2271	. . . . 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 6060	. . . . . 6 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
74		oveq2 6060	. . . . . 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 2249	. . . . 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 2657	. 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 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4111  (class class class)co 6052   CCcc 8130   1c1 8133   + caddc 8135   x. cmul 8137   - cmin 8449   / cdiv 8951   NNcn 9242   2c2 9293   NN0cn0 9501   ZZcz 9582   ^cexp 10907   || cdvds 12481   Primecprime 12812   pCnt cpc 12990   sigma csgm 15898

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252  ax-pre-suploc 8253  ax-addf 8254  ax-mulf 8255

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-disj 4088  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-of 6268  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-2o 6650  df-oadd 6653  df-er 6769  df-map 6886  df-pm 6887  df-en 6978  df-dom 6979  df-fin 6980  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-xnn0 9569  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-xneg 10111  df-xadd 10112  df-ioo 10231  df-ico 10233  df-icc 10234  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-fac 11096  df-bc 11118  df-ihash 11147  df-shft 11508  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-clim 11972  df-sumdc 12047  df-ef 12342  df-e 12343  df-dvds 12482  df-gcd 12658  df-prm 12813  df-pc 12991  df-rest 13475  df-topgen 13494  df-psmet 14740  df-xmet 14741  df-met 14742  df-bl 14743  df-mopn 14744  df-top 14912  df-topon 14925  df-bases 14957  df-ntr 15010  df-cn 15102  df-cnp 15103  df-tx 15167  df-cncf 15485  df-limced 15570  df-dvap 15571  df-relog 15772  df-rpcxp 15773  df-sgm 15899

This theorem is referenced by: (None)