Theorem chtprm 16261

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 9691	. . . . . 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 10042	. . . . 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 16245	. . . 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 16240	. . . . . 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 10740	. . . . . . . 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 6104	. . . . . 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 12157	. . 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 9659	. . . . . . . 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 9268	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A < ( A + 1 ) )
19		zltnle 9701	. . . . . . 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 10449	. . . . . 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 9683	. . . . . . 7 |- 2 e. ZZ
29		zcn 9660	. . . . . . . . . . 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 8278	. . . . . . . . . 10 |- 1 e. CC
32		pncan 8539	. . . . . . . . . 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 12935	. . . . . . . . . . 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 10021	. . . . . . . . . 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 9974	. . . . . . . . 9 |- NN = ( ZZ>= ` 1 )
40		2m1e1 9431	. . . . . . . . . 10 |- ( 2 - 1 ) = 1
41	40	fveq2i 5701	. . . . . . . . 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 10503	. . . . . . 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 10889	. . . 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 10445	. . . . . . . . 9 |- ( x e. ( 2 ... ( A + 1 ) ) -> x e. ZZ )
63		prmdcz 12934	. . . . . . . . 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 7249	. . . 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 12913	. . . . . . 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 10112	. . . . 5 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> p e. RR+ )
78	77	relogcld 16112	. . . 4 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ p e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) -> ( log ` p ) e. RR )
79	78	recnd 8360	. . 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 12199	. 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 10042	. . . . . 6 |- ( A e. ZZ -> A e. QQ )
82	81	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. QQ )
83		chtqval 16245	. . . . 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 16240	. . . . . . 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 10740	. . . . . . . . 9 |- ( A e. ZZ -> ( |_ ` A ) = A )
88	87	adantr 276	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( |_ ` A ) = A )
89	88	oveq2d 6104	. . . . . . 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 12157	. . . 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 12913	. . . . 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 10112	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. RR+ )
97	96	relogcld 16112	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. RR )
98	97	recnd 8360	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( log ` ( A + 1 ) ) e. CC )
99		fveq2 5698	. . . . 5 |- ( p = ( A + 1 ) -> ( log ` p ) = ( log ` ( A + 1 ) ) )
100	99	sumsn 12203	. . . 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 6106	. 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 4131   ` cfv 5380  (class class class)co 6088   Fincfn 7026   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   < clt 8366   <_ cle 8367   - cmin 8504   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   QQcq 10035   [,]cicc 10310   ...cfz 10428   |_cfl 10720   sum_csu 12144   Primecprime 12910   logclog 16085   thetaccht 16233

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305  ax-pre-suploc 8306  ax-addf 8307  ax-mulf 8308

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-inf 7331  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-xneg 10191  df-xadd 10192  df-ioo 10311  df-ico 10313  df-icc 10314  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-e 12441  df-dvds 12580  df-prm 12911  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-ntr 15322  df-cn 15414  df-cnp 15415  df-tx 15479  df-cncf 15797  df-limced 15882  df-dvap 15883  df-relog 16087  df-cht 16236

This theorem is used by:  cht2  16276  cht3  16277