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Theorem ppisval 20341
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.
StepHypRef Expression
1 inss2 3390 . . . . . . . 8  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
2 simpr 447 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
0 [,] A )  i^i  Prime ) )
31, 2sseldi 3178 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  Prime )
4 prmuz2 12776 . . . . . . 7  |-  ( x  e.  Prime  ->  x  e.  ( ZZ>= `  2 )
)
53, 4syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( ZZ>= ` 
2 ) )
6 prmz 12762 . . . . . . . 8  |-  ( x  e.  Prime  ->  x  e.  ZZ )
73, 6syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ZZ )
8 flcl 10927 . . . . . . . 8  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
98adantr 451 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ZZ )
10 inss1 3389 . . . . . . . . . . 11  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1110, 2sseldi 3178 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 0 [,] A ) )
12 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
13 simpl 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
14 elicc2 10715 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1512, 13, 14sylancr 644 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1611, 15mpbid 201 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) )
1716simp3d 969 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  A )
18 flge 10937 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
197, 18syldan 456 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
2017, 19mpbid 201 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  ( |_ `  A ) )
21 eluz2 10236 . . . . . . 7  |-  ( ( |_ `  A )  e.  ( ZZ>= `  x
)  <->  ( x  e.  ZZ  /\  ( |_
`  A )  e.  ZZ  /\  x  <_ 
( |_ `  A
) ) )
227, 9, 20, 21syl3anbrc 1136 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= `  x ) )
23 elfzuzb 10792 . . . . . 6  |-  ( x  e.  ( 2 ... ( |_ `  A
) )  <->  ( x  e.  ( ZZ>= `  2 )  /\  ( |_ `  A
)  e.  ( ZZ>= `  x ) ) )
245, 22, 23sylanbrc 645 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 2 ... ( |_ `  A ) ) )
25 elin 3358 . . . . 5  |-  ( x  e.  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  <->  ( x  e.  ( 2 ... ( |_ `  A ) )  /\  x  e.  Prime ) )
2624, 3, 25sylanbrc 645 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )
2726ex 423 . . 3  |-  ( A  e.  RR  ->  (
x  e.  ( ( 0 [,] A )  i^i  Prime )  ->  x  e.  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) ) )
2827ssrdv 3185 . 2  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 2z 10054 . . . . 5  |-  2  e.  ZZ
30 fzval2 10785 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( |_ `  A )  e.  ZZ )  -> 
( 2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A ) )  i^i 
ZZ ) )
3129, 8, 30sylancr 644 . . . 4  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A
) )  i^i  ZZ ) )
32 inss1 3389 . . . . 5  |-  ( ( 2 [,] ( |_
`  A ) )  i^i  ZZ )  C_  ( 2 [,] ( |_ `  A ) )
3312a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  0  e.  RR )
34 id 19 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR )
35 2re 9815 . . . . . . . 8  |-  2  e.  RR
36 2pos 9828 . . . . . . . 8  |-  0  <  2
3712, 35, 36ltleii 8941 . . . . . . 7  |-  0  <_  2
3837a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  0  <_  2 )
39 flle 10931 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
40 iccss 10718 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
2  /\  ( |_ `  A )  <_  A
) )  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
4133, 34, 38, 39, 40syl22anc 1183 . . . . 5  |-  ( A  e.  RR  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
4232, 41syl5ss 3190 . . . 4  |-  ( A  e.  RR  ->  (
( 2 [,] ( |_ `  A ) )  i^i  ZZ )  C_  ( 0 [,] A
) )
4331, 42eqsstrd 3212 . . 3  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  C_  ( 0 [,] A
) )
44 ssrin 3394 . . 3  |-  ( ( 2 ... ( |_
`  A ) ) 
C_  ( 0 [,] A )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4543, 44syl 15 . 2  |-  ( A  e.  RR  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4628, 45eqssd 3196 1  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737    <_ cle 8868   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   [,]cicc 10659   ...cfz 10782   |_cfl 10924   Primecprime 12758
This theorem is referenced by:  ppisval2  20342  ppifi  20343  ppival2  20366  chtfl  20387  chtprm  20391  chtnprm  20392  ppifl  20398  cht1  20403  chtlepsi  20445  chpval2  20457  chpub  20459
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-icc 10663  df-fz 10783  df-fl 10925  df-dvds 12532  df-prm 12759
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