Theorem ppisval 24830

Description: The set of primes less than A expressed using a finite set of integers. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
ppisval	|- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )

Proof of Theorem ppisval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3834	. . . . . . . 8 |- ( ( 0 [,] A ) i^i Prime ) C_ Prime
2		simpr 477	. . . . . . . 8 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ( ( 0 [,] A ) i^i Prime ) )
3	1, 2	sseldi 3601	. . . . . . 7 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. Prime )
4		prmuz2 15408	. . . . . . 7 |- ( x e. Prime -> x e. ( ZZ>= ` 2 ) )
5	3, 4	syl 17	. . . . . 6 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ( ZZ>= ` 2 ) )
6		prmz 15389	. . . . . . . 8 |- ( x e. Prime -> x e. ZZ )
7	3, 6	syl 17	. . . . . . 7 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ZZ )
8		flcl 12596	. . . . . . . 8 |- ( A e. RR -> ( |_ ` A ) e. ZZ )
9	8	adantr 481	. . . . . . 7 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> ( |_ ` A ) e. ZZ )
10		inss1 3833	. . . . . . . . . . 11 |- ( ( 0 [,] A ) i^i Prime ) C_ ( 0 [,] A )
11	10, 2	sseldi 3601	. . . . . . . . . 10 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ( 0 [,] A ) )
12		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
13		simpl 473	. . . . . . . . . . 11 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> A e. RR )
14		elicc2 12238	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ A e. RR ) -> ( x e. ( 0 [,] A ) <-> ( x e. RR /\ 0 <_ x /\ x <_ A ) ) )
15	12, 13, 14	sylancr 695	. . . . . . . . . 10 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> ( x e. ( 0 [,] A ) <-> ( x e. RR /\ 0 <_ x /\ x <_ A ) ) )
16	11, 15	mpbid 222	. . . . . . . . 9 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> ( x e. RR /\ 0 <_ x /\ x <_ A ) )
17	16	simp3d 1075	. . . . . . . 8 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x <_ A )
18		flge 12606	. . . . . . . . 9 |- ( ( A e. RR /\ x e. ZZ ) -> ( x <_ A <-> x <_ ( |_ ` A ) ) )
19	7, 18	syldan 487	. . . . . . . 8 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> ( x <_ A <-> x <_ ( |_ ` A ) ) )
20	17, 19	mpbid 222	. . . . . . 7 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x <_ ( |_ ` A ) )
21		eluz2 11693	. . . . . . 7 |- ( ( |_ ` A ) e. ( ZZ>= ` x ) <-> ( x e. ZZ /\ ( |_ ` A ) e. ZZ /\ x <_ ( |_ ` A ) ) )
22	7, 9, 20, 21	syl3anbrc 1246	. . . . . 6 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> ( |_ ` A ) e. ( ZZ>= ` x ) )
23		elfzuzb 12336	. . . . . 6 |- ( x e. ( 2 ... ( |_ ` A ) ) <-> ( x e. ( ZZ>= ` 2 ) /\ ( |_ ` A ) e. ( ZZ>= ` x ) ) )
24	5, 22, 23	sylanbrc 698	. . . . 5 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ( 2 ... ( |_ ` A ) ) )
25	24, 3	elind 3798	. . . 4 |- ( ( A e. RR /\ x e. ( ( 0 [,] A ) i^i Prime ) ) -> x e. ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
26	25	ex 450	. . 3 |- ( A e. RR -> ( x e. ( ( 0 [,] A ) i^i Prime ) -> x e. ( ( 2 ... ( |_ ` A ) ) i^i Prime ) ) )
27	26	ssrdv 3609	. 2 |- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) C_ ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
28		2z 11409	. . . . 5 |- 2 e. ZZ
29		fzval2 12329	. . . . 5 |- ( ( 2 e. ZZ /\ ( |_ ` A ) e. ZZ ) -> ( 2 ... ( |_ ` A ) ) = ( ( 2 [,] ( |_ ` A ) ) i^i ZZ ) )
30	28, 8, 29	sylancr 695	. . . 4 |- ( A e. RR -> ( 2 ... ( |_ ` A ) ) = ( ( 2 [,] ( |_ ` A ) ) i^i ZZ ) )
31		inss1 3833	. . . . 5 |- ( ( 2 [,] ( |_ ` A ) ) i^i ZZ ) C_ ( 2 [,] ( |_ ` A ) )
32	12	a1i 11	. . . . . 6 |- ( A e. RR -> 0 e. RR )
33		id 22	. . . . . 6 |- ( A e. RR -> A e. RR )
34		0le2 11111	. . . . . . 7 |- 0 <_ 2
35	34	a1i 11	. . . . . 6 |- ( A e. RR -> 0 <_ 2 )
36		flle 12600	. . . . . 6 |- ( A e. RR -> ( |_ ` A ) <_ A )
37		iccss 12241	. . . . . 6 |- ( ( ( 0 e. RR /\ A e. RR ) /\ ( 0 <_ 2 /\ ( |_ ` A ) <_ A ) ) -> ( 2 [,] ( |_ ` A ) ) C_ ( 0 [,] A ) )
38	32, 33, 35, 36, 37	syl22anc 1327	. . . . 5 |- ( A e. RR -> ( 2 [,] ( |_ ` A ) ) C_ ( 0 [,] A ) )
39	31, 38	syl5ss 3614	. . . 4 |- ( A e. RR -> ( ( 2 [,] ( |_ ` A ) ) i^i ZZ ) C_ ( 0 [,] A ) )
40	30, 39	eqsstrd 3639	. . 3 |- ( A e. RR -> ( 2 ... ( |_ ` A ) ) C_ ( 0 [,] A ) )
41		ssrin 3838	. . 3 |- ( ( 2 ... ( |_ ` A ) ) C_ ( 0 [,] A ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C_ ( ( 0 [,] A ) i^i Prime ) )
42	40, 41	syl 17	. 2 |- ( A e. RR -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) C_ ( ( 0 [,] A ) i^i Prime ) )
43	27, 42	eqssd 3620	1 |- ( A e. RR -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   <_ cle 10075   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   [,]cicc 12178   ...cfz 12326   |_cfl 12591   Primecprime 15385

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

This theorem is referenced by:  ppisval2  24831  ppifi  24832  ppival2  24854  chtfl  24875  chtprm  24879  chtnprm  24880  ppifl  24886  cht1  24891  chtlepsi  24931  chpval2  24943  chpub  24945  chtvalz  30707