Theorem ppinprm 24878

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

Assertion

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

Proof of Theorem ppinprm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3834	. . . . . . . . . . 11 |- ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ Prime
2		simprr 796	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
3	1, 2	sseldi 3601	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. Prime )
4		simprl 794	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. ( A + 1 ) e. Prime )
5		nelne2 2891	. . . . . . . . . 10 |- ( ( x e. Prime /\ -. ( A + 1 ) e. Prime ) -> x =/= ( A + 1 ) )
6	3, 4, 5	syl2anc 693	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x =/= ( A + 1 ) )
7		velsn 4193	. . . . . . . . . 10 |- ( x e. { ( A + 1 ) } <-> x = ( A + 1 ) )
8	7	necon3bbii 2841	. . . . . . . . 9 |- ( -. x e. { ( A + 1 ) } <-> x =/= ( A + 1 ) )
9	6, 8	sylibr 224	. . . . . . . 8 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. x e. { ( A + 1 ) } )
10		inss1 3833	. . . . . . . . . . . 12 |- ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( 2 ... ( A + 1 ) )
11	10, 2	sseldi 3601	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... ( A + 1 ) ) )
12		2z 11409	. . . . . . . . . . . 12 |- 2 e. ZZ
13		zcn 11382	. . . . . . . . . . . . . . . 16 |- ( A e. ZZ -> A e. CC )
14	13	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. CC )
15		ax-1cn 9994	. . . . . . . . . . . . . . 15 |- 1 e. CC
16		pncan 10287	. . . . . . . . . . . . . . 15 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
17	14, 15, 16	sylancl 694	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) = A )
18		elfzuz2 12346	. . . . . . . . . . . . . . 15 |- ( x e. ( 2 ... ( A + 1 ) ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
19		uz2m1nn 11763	. . . . . . . . . . . . . . 15 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
20	11, 18, 19	3syl 18	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) e. NN )
21	17, 20	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. NN )
22		nnuz 11723	. . . . . . . . . . . . . 14 |- NN = ( ZZ>= ` 1 )
23		2m1e1 11135	. . . . . . . . . . . . . . 15 |- ( 2 - 1 ) = 1
24	23	fveq2i 6194	. . . . . . . . . . . . . 14 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
25	22, 24	eqtr4i 2647	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` ( 2 - 1 ) )
26	21, 25	syl6eleq 2711	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
27		fzsuc2 12398	. . . . . . . . . . . 12 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
28	12, 26, 27	sylancr 695	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
29	11, 28	eleqtrd 2703	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) )
30		elun 3753	. . . . . . . . . 10 |- ( x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) <-> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
31	29, 30	sylib 208	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
32	31	ord 392	. . . . . . . 8 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( -. x e. ( 2 ... A ) -> x e. { ( A + 1 ) } ) )
33	9, 32	mt3d 140	. . . . . . 7 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... A ) )
34	33, 3	elind 3798	. . . . . 6 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) i^i Prime ) )
35	34	expr 643	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) -> x e. ( ( 2 ... A ) i^i Prime ) ) )
36	35	ssrdv 3609	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( ( 2 ... A ) i^i Prime ) )
37		uzid 11702	. . . . . 6 |- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
38	37	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` A ) )
39		peano2uz 11741	. . . . 5 |- ( A e. ( ZZ>= ` A ) -> ( A + 1 ) e. ( ZZ>= ` A ) )
40		fzss2 12381	. . . . 5 |- ( ( A + 1 ) e. ( ZZ>= ` A ) -> ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) )
41		ssrin 3838	. . . . 5 |- ( ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
42	38, 39, 40, 41	4syl 19	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
43	36, 42	eqssd 3620	. . 3 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
44	43	fveq2d 6195	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) = ( # ` ( ( 2 ... A ) i^i Prime ) ) )
45		peano2z 11418	. . . 4 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
46	45	adantr 481	. . 3 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
47		ppival2 24854	. . 3 |- ( ( A + 1 ) e. ZZ -> (π ` ( A + 1 ) ) = ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
48	46, 47	syl 17	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = ( # ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
49		ppival2 24854	. . 3 |- ( A e. ZZ -> (π ` A ) = ( # ` ( ( 2 ... A ) i^i Prime ) ) )
50	49	adantr 481	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` A ) = ( # ` ( ( 2 ... A ) i^i Prime ) ) )
51	44, 48, 50	3eqtr4d 2666	1 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = (π ` A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   - 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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-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  ppi2i  24895  bposlem5  25013