Theorem ppiprm 16170

Description: The prime-counting function π at a prime. (Contributed by Mario Carneiro, 19-Sep-2014.)

Assertion

Ref	Expression
ppiprm	|- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( (π ` A ) + 1 ) )

Proof of Theorem ppiprm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2z 9676	. . . . . 6 |- 2 e. ZZ
2	1	a1i 9	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> 2 e. ZZ )
3		simpl 109	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. ZZ )
4	2, 3	fzfigd 10881	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... A ) e. Fin )
5		inss1 3451	. . . . 5 |- ( ( 2 ... A ) i^i Prime ) C_ ( 2 ... A )
6	5	a1i 9	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) C_ ( 2 ... A ) )
7		orc 724	. . . . . . . . 9 |- ( x e. ( 2 ... A ) -> ( x e. ( 2 ... A ) \/ -. x e. ( 2 ... A ) ) )
8		df-dc 847	. . . . . . . . 9 |- (DECID x e. ( 2 ... A ) <-> ( x e. ( 2 ... A ) \/ -. x e. ( 2 ... A ) ) )
9	7, 8	sylibr 134	. . . . . . . 8 |- ( x e. ( 2 ... A ) -> DECID x e. ( 2 ... A ) )
10	9	adantl 277	. . . . . . 7 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... A ) ) -> DECID x e. ( 2 ... A ) )
11		elfzelz 10438	. . . . . . . . 9 |- ( x e. ( 2 ... A ) -> x e. ZZ )
12		prmdcz 12925	. . . . . . . . 9 |- ( x e. ZZ -> DECID x e. Prime )
13	11, 12	syl 14	. . . . . . . 8 |- ( x e. ( 2 ... A ) -> DECID x e. Prime )
14	13	adantl 277	. . . . . . 7 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... A ) ) -> DECID x e. Prime )
15	10, 14	dcand 945	. . . . . 6 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... A ) ) -> DECID ( x e. ( 2 ... A ) /\ x e. Prime ) )
16		elin 3412	. . . . . . 7 |- ( x e. ( ( 2 ... A ) i^i Prime ) <-> ( x e. ( 2 ... A ) /\ x e. Prime ) )
17	16	dcbii 852	. . . . . 6 |- (DECID x e. ( ( 2 ... A ) i^i Prime ) <-> DECID ( x e. ( 2 ... A ) /\ x e. Prime ) )
18	15, 17	sylibr 134	. . . . 5 |- ( ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) /\ x e. ( 2 ... A ) ) -> DECID x e. ( ( 2 ... A ) i^i Prime ) )
19	18	ralrimiva 2623	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A. x e. ( 2 ... A )DECID x e. ( ( 2 ... A ) i^i Prime ) )
20		ssfidc 7245	. . . 4 |- ( ( ( 2 ... A ) e. Fin /\ ( ( 2 ... A ) i^i Prime ) C_ ( 2 ... A ) /\ A. x e. ( 2 ... A )DECID x e. ( ( 2 ... A ) i^i Prime ) ) -> ( ( 2 ... A ) i^i Prime ) e. Fin )
21	4, 6, 19, 20	syl3anc 1278	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) e. Fin )
22		zre 9652	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
23	22	adantr 276	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. RR )
24	23	ltp1d 9262	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A < ( A + 1 ) )
25		peano2z 9684	. . . . . . 7 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
26	25	adantr 276	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
27		zltnle 9694	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. ZZ ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) )
28	26, 27	syldan 282	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) )
29	24, 28	mpbid 147	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) <_ A )
30		elinel1 3415	. . . . 5 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) e. ( 2 ... A ) )
31		elfzle2 10442	. . . . 5 |- ( ( A + 1 ) e. ( 2 ... A ) -> ( A + 1 ) <_ A )
32	30, 31	syl 14	. . . 4 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) <_ A )
33	29, 32	nsyl 637	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) )
34		hashunsng 11262	. . . . 5 |- ( ( A + 1 ) e. ZZ -> ( ( ( ( 2 ... A ) i^i Prime ) e. Fin /\ -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) ) -> (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) = ( (♯ ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) ) )
35	25, 34	syl 14	. . . 4 |- ( A e. ZZ -> ( ( ( ( 2 ... A ) i^i Prime ) e. Fin /\ -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) ) -> (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) = ( (♯ ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) ) )
36	35	adantr 276	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( ( ( 2 ... A ) i^i Prime ) e. Fin /\ -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) ) -> (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) = ( (♯ ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) ) )
37	21, 33, 36	mp2and 437	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) = ( (♯ ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) )
38		ppival2 16161	. . . 4 |- ( ( A + 1 ) e. ZZ -> (π ` ( A + 1 ) ) = (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
39	26, 38	syl 14	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
40		zcn 9653	. . . . . . . . . . . 12 |- ( A e. ZZ -> A e. CC )
41	40	adantr 276	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. CC )
42		ax-1cn 8272	. . . . . . . . . . 11 |- 1 e. CC
43		pncan 8533	. . . . . . . . . . 11 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
44	41, 42, 43	sylancl 417	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) = A )
45		prmuz2 12926	. . . . . . . . . . . 12 |- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
46	45	adantl 277	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
47		uz2m1nn 10014	. . . . . . . . . . 11 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
48	46, 47	syl 14	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) e. NN )
49	44, 48	eqeltrrd 2316	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. NN )
50		nnuz 9967	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
51		2m1e1 9424	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
52	51	fveq2i 5698	. . . . . . . . . 10 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
53	50, 52	eqtr4i 2262	. . . . . . . . 9 |- NN = ( ZZ>= ` ( 2 - 1 ) )
54	49, 53	eleqtrdi 2331	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
55		fzsuc2 10496	. . . . . . . 8 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
56	1, 54, 55	sylancr 418	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
57	56	ineq1d 3431	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) )
58		indir 3480	. . . . . 6 |- ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) )
59	57, 58	eqtrdi 2287	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) ) )
60		simpr 110	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. Prime )
61	60	snssd 3860	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> { ( A + 1 ) } C_ Prime )
62		dfss2 3237	. . . . . . 7 |- ( { ( A + 1 ) } C_ Prime <-> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
63	61, 62	sylib 122	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
64	63	uneq2d 3383	. . . . 5 |- ( ( 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 ) } ) )
65	59, 64	eqtrd 2271	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) )
66	65	fveq2d 5699	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) = (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) )
67	39, 66	eqtrd 2271	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = (♯ ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) )
68		ppival2 16161	. . . 4 |- ( A e. ZZ -> (π ` A ) = (♯ ` ( ( 2 ... A ) i^i Prime ) ) )
69	68	adantr 276	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` A ) = (♯ ` ( ( 2 ... A ) i^i Prime ) ) )
70	69	oveq1d 6100	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( (π ` A ) + 1 ) = ( (♯ ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) )
71	37, 67, 70	3eqtr4d 2281	1 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( (π ` 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   {csn 3709   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Fincfn 7022   CCcc 8177   RRcr 8178   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8498   NNcn 9306   2c2 9357   ZZcz 9648   ZZ>=cuz 9930   ...cfz 10421  ♯chash 11228   Primecprime 12901  πcppi 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

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-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-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  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-en 7023  df-dom 7024  df-fin 7025  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-icc 10307  df-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-ihash 11229  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902  df-ppi 16154

This theorem is used by:  ppiqp1le  16173  ppi1i  16177  bposlem5  16213