Theorem ppinprm 16260

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		simprr 537	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
2	1	elin2d 3419	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. Prime )
3		simprl 535	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. ( A + 1 ) e. Prime )
4		nelne2 2511	. . . . . . . . . 10 |- ( ( x e. Prime /\ -. ( A + 1 ) e. Prime ) -> x =/= ( A + 1 ) )
5	2, 3, 4	syl2anc 415	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x =/= ( A + 1 ) )
6		velsn 3726	. . . . . . . . . 10 |- ( x e. { ( A + 1 ) } <-> x = ( A + 1 ) )
7	6	necon3bbii 2457	. . . . . . . . 9 |- ( -. x e. { ( A + 1 ) } <-> x =/= ( A + 1 ) )
8	5, 7	sylibr 134	. . . . . . . 8 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. x e. { ( A + 1 ) } )
9	1	elin1d 3418	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... ( A + 1 ) ) )
10		2z 9683	. . . . . . . . . . 11 |- 2 e. ZZ
11		zcn 9660	. . . . . . . . . . . . . . 15 |- ( A e. ZZ -> A e. CC )
12	11	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. CC )
13		ax-1cn 8278	. . . . . . . . . . . . . 14 |- 1 e. CC
14		pncan 8539	. . . . . . . . . . . . . 14 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
15	12, 13, 14	sylancl 417	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) = A )
16		elfzuz2 10450	. . . . . . . . . . . . . 14 |- ( x e. ( 2 ... ( A + 1 ) ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
17		uz2m1nn 10021	. . . . . . . . . . . . . 14 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
18	9, 16, 17	3syl 17	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) e. NN )
19	15, 18	eqeltrrd 2316	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. NN )
20		nnuz 9974	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
21		2m1e1 9431	. . . . . . . . . . . . . 14 |- ( 2 - 1 ) = 1
22	21	fveq2i 5701	. . . . . . . . . . . . 13 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
23	20, 22	eqtr4i 2262	. . . . . . . . . . . 12 |- NN = ( ZZ>= ` ( 2 - 1 ) )
24	19, 23	eleqtrdi 2331	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
25		fzsuc2 10503	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
26	10, 24, 25	sylancr 418	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
27	9, 26	eleqtrd 2317	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) )
28		elun 3370	. . . . . . . . 9 |- ( x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) <-> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
29	27, 28	sylib 122	. . . . . . . 8 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
30	8, 29	ecased 1390	. . . . . . 7 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... A ) )
31	30, 2	elind 3414	. . . . . 6 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) i^i Prime ) )
32	31	expr 375	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) -> x e. ( ( 2 ... A ) i^i Prime ) ) )
33	32	ssrdv 3254	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( ( 2 ... A ) i^i Prime ) )
34		uzid 9952	. . . . . 6 |- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
35	34	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` A ) )
36		peano2uz 9999	. . . . 5 |- ( A e. ( ZZ>= ` A ) -> ( A + 1 ) e. ( ZZ>= ` A ) )
37		fzss2 10487	. . . . 5 |- ( ( A + 1 ) e. ( ZZ>= ` A ) -> ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) )
38		ssrin 3456	. . . . 5 |- ( ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
39	35, 36, 37, 38	4syl 18	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
40	33, 39	eqssd 3265	. . 3 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
41	40	fveq2d 5702	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) = (♯ ` ( ( 2 ... A ) i^i Prime ) ) )
42		peano2z 9691	. . . 4 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
43	42	adantr 276	. . 3 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
44		ppival2 16249	. . 3 |- ( ( A + 1 ) e. ZZ -> (π ` ( A + 1 ) ) = (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
45	43, 44	syl 14	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = (♯ ` ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) )
46		ppival2 16249	. . 3 |- ( A e. ZZ -> (π ` A ) = (♯ ` ( ( 2 ... A ) i^i Prime ) ) )
47	46	adantr 276	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` A ) = (♯ ` ( ( 2 ... A ) i^i Prime ) ) )
48	41, 45, 47	3eqtr4d 2281	1 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> (π ` ( A + 1 ) ) = (π ` A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   i^i cin 3219   C_ wss 3220   {csn 3709   ` cfv 5380  (class class class)co 6088   CCcc 8183   1c1 8186   + caddc 8188   - cmin 8504   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428  ♯chash 11236   Primecprime 12910  πcppi 16234

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

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 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-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  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-icc 10314  df-fz 10429  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-prm 12911  df-ppi 16237

This theorem is used by:  ppiqp1le  16267  ppi2i  16273  bposlem5  16315