Theorem perfect 15640

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 15636. 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 12615	. . . . . . . 8 |- 2 e. Prime
3		simpll 527	. . . . . . . 8 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. NN )
4		pcelnn 12810	. . . . . . . 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 9536	. . . . 5 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. ZZ )
8	7	peano2zd 9540	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) + 1 ) e. ZZ )
9		pcdvds 12804	. . . . . . . . 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 9240	. . . . . . . . . 10 |- 2 e. NN
12	6	nnnn0d 9390	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN0 )
13		nnexpcl 10741	. . . . . . . . . 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 12273	. . . . . . . . 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 12808	. . . . . . . 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 9086	. . . . . . . . . . 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 9092	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. CC )
24	14	nnap0d 9124	. . . . . . . . . 10 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) # 0 )
25	22, 23, 24	divcanap2d 8907	. . . . . . . . 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 5990	. . . . . . . 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 5990	. . . . . . . 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 2252	. . . . . . 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 15639	. . . . . 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 2287	. . . 4 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime )
33	6	nncnd 9092	. . . . . . . . 9 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. CC )
34		ax-1cn 8060	. . . . . . . . 9 |- 1 e. CC
35		pncan 8320	. . . . . . . . 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 2215	. . . . . . 7 |- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
38	37	oveq2d 5990	. . . . . 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 5992	. . . . 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 2244	. . . 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 5982	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ p ) = ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) )
42	41	oveq1d 5989	. . . . . . 7 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ p ) - 1 ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) )
43	42	eleq1d 2278	. . . . . 6 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( 2 ^ p ) - 1 ) e. Prime <-> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) )
44		oveq1 5981	. . . . . . . . 9 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( p - 1 ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) )
45	44	oveq2d 5990	. . . . . . . 8 |- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ ( p - 1 ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) )
46	45, 42	oveq12d 5992	. . . . . . 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 2221	. . . . . 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 2887	. . . 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 1250	. . 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 15637	. . . . . 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 9149	. . . . . . . . 9 |- 2 e. CC
54		mersenne 15636	. . . . . . . . . 10 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. Prime )
55		prmnn 12598	. . . . . . . . . 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 10756	. . . . . . . . 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 9378	. . . . . . . . . . 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 10746	. . . . . . . . . 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 8096	. . . . . . . . 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 2242	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) )
66	65	oveq1d 5989	. . . . . 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 9151	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> 2 e. CC )
68		prmnn 12598	. . . . . . . . 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 9092	. . . . . . 7 |- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. CC )
71	67, 62, 70	mulassd 8138	. . . . . 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 2246	. . . . 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 5982	. . . . . 6 |- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) )
74		oveq2 5982	. . . . . 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 2224	. . . . 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 2623	. 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 1375   e. wcel 2180   E.wrex 2489   class class class wbr 4062  (class class class)co 5974   CCcc 7965   1c1 7968   + caddc 7970   x. cmul 7972   - cmin 8285   / cdiv 8787   NNcn 9078   2c2 9129   NN0cn0 9337   ZZcz 9414   ^cexp 10727   || cdvds 12264   Primecprime 12595   pCnt cpc 12773   sigma csgm 15620

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087  ax-pre-suploc 8088  ax-addf 8089  ax-mulf 8090

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-tp 3654  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-disj 4039  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-2o 6533  df-oadd 6536  df-er 6650  df-map 6767  df-pm 6768  df-en 6858  df-dom 6859  df-fin 6860  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-xnn0 9401  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-xneg 9936  df-xadd 9937  df-ioo 10056  df-ico 10058  df-icc 10059  df-fz 10173  df-fzo 10307  df-fl 10457  df-mod 10512  df-seqfrec 10637  df-exp 10728  df-fac 10915  df-bc 10937  df-ihash 10965  df-shft 11292  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-sumdc 11831  df-ef 12125  df-e 12126  df-dvds 12265  df-gcd 12441  df-prm 12596  df-pc 12774  df-rest 13240  df-topgen 13259  df-psmet 14472  df-xmet 14473  df-met 14474  df-bl 14475  df-mopn 14476  df-top 14637  df-topon 14650  df-bases 14682  df-ntr 14735  df-cn 14827  df-cnp 14828  df-tx 14892  df-cncf 15210  df-limced 15295  df-dvap 15296  df-relog 15497  df-rpcxp 15498  df-sgm 15621

This theorem is referenced by: (None)