Theorem chtprm 16180

Description: The Chebyshev function at a prime. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
chtprm	|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = ( ( theta ` A ) + ( log ` ( A + 1 ) ) ) )

Proof of Theorem chtprm

Dummy variables p x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2z 9684	. . . . . 6 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
2	1	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
3		zq 10035	. . . . 5 |- ( ( A + 1 ) e. ZZ -> ( A + 1 ) e. QQ )
4	2, 3	syl 14	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. QQ )
5		chtqval 16164	. . . 4 |- ( ( A + 1 ) e. QQ -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) )
6	4, 5	syl 14	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) )
7		ppiqsval 16159	. . . . . 6 |- ( ( A + 1 ) e. QQ -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) )
8	4, 7	syl 14	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) )
9		flid 10732	. . . . . . . 8 |- ( ( A + 1 ) e. ZZ -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) )
10	2, 9	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) )
11	10	oveq2d 6101	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` ( A + 1 ) ) ) = ( 2 ... ( A + 1 ) ) )
12	11	ineq1d 3431	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) = ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
13	8, 12	eqtrd 2271	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
14	13	sumeq1d 12148	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) )
15	6, 14	eqtrd 2271	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) )
16		zre 9652	. . . . . . . 8 |- ( A e. ZZ -> A e. RR )
17	16	adantr 276	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. RR )
18	17	ltp1d 9262	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A < ( A + 1 ) )
19		zltnle 9694	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. ZZ ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) )
20	2, 19	syldan 282	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) )
21	18, 20	mpbid 147	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) <_ A )
22		elinel1 3415	. . . . . 6 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) e. ( 2 ... A ) )
23		elfzle2 10442	. . . . . 6 |- ( ( A + 1 ) e. ( 2 ... A ) -> ( A + 1 ) <_ A )
24	22, 23	syl 14	. . . . 5 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) <_ A )
25	21, 24	nsyl 637	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) )
26		disjsn 3771	. . . 4 |- ( ( ( ( 2 ... A ) i^i Prime ) i^i { ( A + 1 ) } ) = (/) <-> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) )
27	25, 26	sylibr 134	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( ( 2 ... A ) i^i Prime ) i^i { ( A + 1 ) } ) = (/) )
28		2z 9676	. . . . . . 7 |- 2 e. ZZ
29		zcn 9653	. . . . . . . . . . 11 |- ( A e. ZZ -> A e. CC )
30	29	adantr 276	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. CC )
31		ax-1cn 8272	. . . . . . . . . 10 |- 1 e. CC
32		pncan 8533	. . . . . . . . . 10 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
33	30, 31, 32	sylancl 417	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) = A )
34		prmuz2 12926	. . . . . . . . . . 11 |- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
35	34	adantl 277	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
36		uz2m1nn 10014	. . . . . . . . . 10 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
37	35, 36	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) e. NN )
38	33, 37	eqeltrrd 2316	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. NN )
39		nnuz 9967	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
40		2m1e1 9424	. . . . . . . . . 10 |- ( 2 - 1 ) = 1
41	40	fveq2i 5698	. . . . . . . . 9 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
42	39, 41	eqtr4i 2262	. . . . . . . 8 |- NN = ( ZZ>= ` ( 2 - 1 ) )
43	38, 42	eleqtrdi 2331	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
44		fzsuc2 10496	. . . . . . 7 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
45	28, 43, 44	sylancr 418	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
46	45	ineq1d 3431	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) )
47		indir 3480	. . . . 5 |- ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) )
48	46, 47	eqtrdi 2287	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) )
49		simpr 110	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. Prime )
50	49	snssd 3860	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> { ( A + 1 ) } C_ Prime )
51		dfss2 3237	. . . . . 6 |- ( { ( A + 1 ) } C_ Prime <-> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
52	50, 51	sylib 122	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
53	52	uneq2d 3383	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) )
54	48, 53	eqtrd 2271	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) )
55	28	a1i 9	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> 2 e. ZZ )
56	55, 2	fzfigd 10881	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) e. Fin )
57		inss1 3451	. . . . 5 |- ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) )
58	57	a1i 9	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) ) )
59		animorrl 838	. . . . . . . 8 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... ( A + 1 ) ) ) -> ( x e. ( 2 ... ( A + 1 ) ) \/ -. x e. ( 2 ... ( A + 1 ) ) ) )
60		df-dc 847	. . . . . . . 8 |- (DECID x e. ( 2 ... ( A + 1 ) ) <-> ( x e. ( 2 ... ( A + 1 ) ) \/ -. x e. ( 2 ... ( A + 1 ) ) ) )
61	59, 60	sylibr 134	. . . . . . 7 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... ( A + 1 ) ) ) -> DECID x e. ( 2 ... ( A + 1 ) ) )
62		elfzelz 10438	. . . . . . . . 9 |- ( x e. ( 2 ... ( A + 1 ) ) -> x e. ZZ )
63		prmdcz 12925	. . . . . . . . 9 |- ( x e. ZZ -> DECID x e. Prime )
64	62, 63	syl 14	. . . . . . . 8 |- ( x e. ( 2 ... ( A + 1 ) ) -> DECID x e. Prime )
65	64	adantl 277	. . . . . . 7 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... ( A + 1 ) ) ) -> DECID x e. Prime )
66	61, 65	dcand 945	. . . . . 6 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... ( A + 1 ) ) ) -> DECID ( x e. ( 2 ... ( A + 1 ) ) /\ x e. Prime ) )
67		elin 3412	. . . . . . 7 |- ( x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) <-> ( x e. ( 2 ... ( A + 1 ) ) /\ x e. Prime ) )
68	67	dcbii 852	. . . . . 6 |- (DECID x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) <-> DECID ( x e. ( 2 ... ( A + 1 ) ) /\ x e. Prime ) )
69	66, 68	sylibr 134	. . . . 5 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... ( A + 1 ) ) ) -> DECID x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
70	69	ralrimiva 2623	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A. x e. ( 2 ... ( A + 1 ) )DECID x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
71		ssfidc 7245	. . . 4 |- ( ( ( 2 ... ( A + 1 ) ) e. Fin /\ ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) ) /\ A. x e. ( 2 ... ( A + 1 ) )DECID x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) e. Fin )
72	56, 58, 70, 71	syl3anc 1278	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) e. Fin )
73		simpr 110	. . . . . . . 8 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
74	73	elin2d 3419	. . . . . . 7 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. Prime )
75		prmnn 12904	. . . . . . 7 |- ( p e. Prime -> p e. NN )
76	74, 75	syl 14	. . . . . 6 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. NN )
77	76	nnrpd 10105	. . . . 5 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. RR+ )
78	77	relogcld 16034	. . . 4 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( log ` p ) e. RR )
79	78	recnd 8354	. . 3 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( log ` p ) e. CC )
80	27, 54, 72, 79	fsumsplit 12190	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ( log ` p ) = ( sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) + sum_ p e. { ( A + 1 ) } ( log ` p ) ) )
81		zq 10035	. . . . . 6 |- ( A e. ZZ -> A e. QQ )
82	81	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. QQ )
83		chtqval 16164	. . . . 5 |- ( A e. QQ -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
84	82, 83	syl 14	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
85		ppiqsval 16159	. . . . . . 7 |- ( A e. QQ -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
86	82, 85	syl 14	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
87		flid 10732	. . . . . . . . 9 |- ( A e. ZZ -> ( |_ ` A ) = A )
88	87	adantr 276	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( |_ ` A ) = A )
89	88	oveq2d 6101	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` A ) ) = ( 2 ... A ) )
90	89	ineq1d 3431	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
91	86, 90	eqtrd 2271	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
92	91	sumeq1d 12148	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) )
93	84, 92	eqtr2d 2272	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) = ( theta ` A ) )
94		prmnn 12904	. . . . 5 |- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. NN )
95	94	adantl 277	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. NN )
96	95	nnrpd 10105	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. RR+ )
97	96	relogcld 16034	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. RR )
98	97	recnd 8354	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. CC )
99		fveq2 5695	. . . . 5 |- ( p = ( A + 1 ) -> ( log ` p ) = ( log ` ( A + 1 ) ) )
100	99	sumsn 12194	. . . 4 |- ( ( ( A + 1 ) e. NN /\ ( log ` ( A + 1 ) ) e. CC ) -> sum_ p e. { ( A + 1 ) } ( log ` p ) = ( log ` ( A + 1 ) ) )
101	95, 98, 100	syl2anc 415	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> sum_ p e. { ( A + 1 ) } ( log ` p ) = ( log ` ( A + 1 ) ) )
102	93, 101	oveq12d 6103	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( sum_ p e. ( ( 2 ... A ) i^i Prime ) ( log ` p ) + sum_ p e. { ( A + 1 ) } ( log ` p ) ) = ( ( theta ` A ) + ( log ` ( A + 1 ) ) ) )
103	15, 80, 102	3eqtrd 2275	1 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = ( ( theta ` A ) + ( log ` ( A + 1 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3709   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Fincfn 7022   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8498   NNcn 9306   2c2 9357   ZZcz 9648   ZZ>=cuz 9930   QQcq 10028   [,]cicc 10303   ...cfz 10421   |_cfl 10713   sum_csu 12135   Primecprime 12901   logclog 16007   thetaccht 16152

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  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

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-disj 4107  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-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-xneg 10184  df-xadd 10185  df-ioo 10304  df-ico 10306  df-icc 10307  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-fac 11178  df-bc 11200  df-ihash 11229  df-shft 11594  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-ef 12431  df-e 12432  df-dvds 12571  df-prm 12902  df-rest 13644  df-topgen 13663  df-psmet 14929  df-xmet 14930  df-met 14931  df-bl 14932  df-mopn 14933  df-top 15148  df-topon 15161  df-bases 15193  df-ntr 15246  df-cn 15338  df-cnp 15339  df-tx 15403  df-cncf 15721  df-limced 15806  df-dvap 15807  df-relog 16009  df-cht 16155

This theorem is used by:  cht2  16195  cht3  16196