Theorem ppiprm 24877

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

Step	Hyp	Ref	Expression
1		fzfid 12772	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... A ) e. Fin )
2		inss1 3833	. . . 4 |- ( ( 2 ... A ) i^i Prime ) C_ ( 2 ... A )
3		ssfi 8180	. . . 4 |- ( ( ( 2 ... A ) e. Fin /\ ( ( 2 ... A ) i^i Prime ) C_ ( 2 ... A ) ) -> ( ( 2 ... A ) i^i Prime ) e. Fin )
4	1, 2, 3	sylancl 694	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) e. Fin )
5		zre 11381	. . . . . . 7 |- ( A e. ZZ -> A e. RR )
6	5	adantr 481	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. RR )
7	6	ltp1d 10954	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A < ( A + 1 ) )
8		peano2z 11418	. . . . . . . 8 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
9	8	adantr 481	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
10	9	zred 11482	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. RR )
11	6, 10	ltnled 10184	. . . . 5 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A < ( A + 1 ) <-> -. ( A + 1 ) <_ A ) )
12	7, 11	mpbid 222	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) <_ A )
13	2	sseli 3599	. . . . 5 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) e. ( 2 ... A ) )
14		elfzle2 12345	. . . . 5 |- ( ( A + 1 ) e. ( 2 ... A ) -> ( A + 1 ) <_ A )
15	13, 14	syl 17	. . . 4 |- ( ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) -> ( A + 1 ) <_ A )
16	12, 15	nsyl 135	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> -. ( A + 1 ) e. ( ( 2 ... A ) i^i Prime ) )
17		ovex 6678	. . . 4 |- ( A + 1 ) e. _V
18		hashunsng 13181	. . . 4 |- ( ( A + 1 ) e. _V -> ( ( ( ( 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 ) ) )
19	17, 18	ax-mp 5	. . 3 |- ( ( ( ( 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 ) )
20	4, 16, 19	syl2anc 693	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( # ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) = ( ( # ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) )
21		ppival2 24854	. . . 4 |- ( ( A + 1 ) e. ZZ -> (π ` ( A + 1 ) ) = ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
22	9, 21	syl 17	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
23		2z 11409	. . . . . . . 8 |- 2 e. ZZ
24		zcn 11382	. . . . . . . . . . . 12 |- ( A e. ZZ -> A e. CC )
25	24	adantr 481	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. CC )
26		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
27		pncan 10287	. . . . . . . . . . 11 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
28	25, 26, 27	sylancl 694	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) = A )
29		prmuz2 15408	. . . . . . . . . . . 12 |- ( ( A + 1 ) e. Prime -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
30	29	adantl 482	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
31		uz2m1nn 11763	. . . . . . . . . . 11 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
32	30, 31	syl 17	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( A + 1 ) - 1 ) e. NN )
33	28, 32	eqeltrrd 2702	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. NN )
34		nnuz 11723	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
35		2m1e1 11135	. . . . . . . . . . 11 |- ( 2 - 1 ) = 1
36	35	fveq2i 6194	. . . . . . . . . 10 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
37	34, 36	eqtr4i 2647	. . . . . . . . 9 |- NN = ( ZZ>= ` ( 2 - 1 ) )
38	33, 37	syl6eleq 2711	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
39		fzsuc2 12398	. . . . . . . 8 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
40	23, 38, 39	sylancr 695	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
41	40	ineq1d 3813	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) )
42		indir 3875	. . . . . 6 |- ( ( ( 2 ... A ) u. { ( A + 1 ) } ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. ( { ( A + 1 ) } i^i Prime ) )
43	41, 42	syl6eq 2672	. . . . 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 ) ) )
44		simpr 477	. . . . . . . 8 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( A + 1 ) e. Prime )
45	44	snssd 4340	. . . . . . 7 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> { ( A + 1 ) } C_ Prime )
46		df-ss 3588	. . . . . . 7 |- ( { ( A + 1 ) } C_ Prime <-> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
47	45, 46	sylib 208	. . . . . 6 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( { ( A + 1 ) } i^i Prime ) = { ( A + 1 ) } )
48	47	uneq2d 3767	. . . . 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 ) } ) )
49	43, 48	eqtrd 2656	. . . 4 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) )
50	49	fveq2d 6195	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) = ( # ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) )
51	22, 50	eqtrd 2656	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( # ` ( ( ( 2 ... A ) i^i Prime ) u. { ( A + 1 ) } ) ) )
52		ppival2 24854	. . . 4 |- ( A e. ZZ -> (π ` A ) = ( # ` ( ( 2 ... A ) i^i Prime ) ) )
53	52	adantr 481	. . 3 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` A ) = ( # ` ( ( 2 ... A ) i^i Prime ) ) )
54	53	oveq1d 6665	. 2 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> ( (π ` A ) + 1 ) = ( ( # ` ( ( 2 ... A ) i^i Prime ) ) + 1 ) )
55	20, 51, 54	3eqtr4d 2666	1 |- ( ( A e. ZZ /\ ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( (π ` A ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117   Primecprime 15385  πcppi 24820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-ppi 24826

This theorem is referenced by:  ppip1le  24887  ppi1i  24894  bposlem5  25013