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Theorem ppisval 20886
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 3562 . . . . . . . 8  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
2 simpr 448 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
0 [,] A )  i^i  Prime ) )
31, 2sseldi 3346 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  Prime )
4 prmuz2 13097 . . . . . . 7  |-  ( x  e.  Prime  ->  x  e.  ( ZZ>= `  2 )
)
53, 4syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( ZZ>= ` 
2 ) )
6 prmz 13083 . . . . . . . 8  |-  ( x  e.  Prime  ->  x  e.  ZZ )
73, 6syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ZZ )
8 flcl 11204 . . . . . . . 8  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
98adantr 452 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ZZ )
10 inss1 3561 . . . . . . . . . . 11  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1110, 2sseldi 3346 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 0 [,] A ) )
12 0re 9091 . . . . . . . . . . 11  |-  0  e.  RR
13 simpl 444 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
14 elicc2 10975 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1512, 13, 14sylancr 645 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1611, 15mpbid 202 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) )
1716simp3d 971 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  A )
18 flge 11214 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
197, 18syldan 457 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
2017, 19mpbid 202 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  ( |_ `  A ) )
21 eluz2 10494 . . . . . . 7  |-  ( ( |_ `  A )  e.  ( ZZ>= `  x
)  <->  ( x  e.  ZZ  /\  ( |_
`  A )  e.  ZZ  /\  x  <_ 
( |_ `  A
) ) )
227, 9, 20, 21syl3anbrc 1138 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= `  x ) )
23 elfzuzb 11053 . . . . . 6  |-  ( x  e.  ( 2 ... ( |_ `  A
) )  <->  ( x  e.  ( ZZ>= `  2 )  /\  ( |_ `  A
)  e.  ( ZZ>= `  x ) ) )
245, 22, 23sylanbrc 646 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 2 ... ( |_ `  A ) ) )
25 elin 3530 . . . . 5  |-  ( x  e.  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime )  <->  ( x  e.  ( 2 ... ( |_ `  A ) )  /\  x  e.  Prime ) )
2624, 3, 25sylanbrc 646 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )
2726ex 424 . . 3  |-  ( A  e.  RR  ->  (
x  e.  ( ( 0 [,] A )  i^i  Prime )  ->  x  e.  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) ) )
2827ssrdv 3354 . 2  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 2z 10312 . . . . 5  |-  2  e.  ZZ
30 fzval2 11046 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( |_ `  A )  e.  ZZ )  -> 
( 2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A ) )  i^i 
ZZ ) )
3129, 8, 30sylancr 645 . . . 4  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A
) )  i^i  ZZ ) )
32 inss1 3561 . . . . 5  |-  ( ( 2 [,] ( |_
`  A ) )  i^i  ZZ )  C_  ( 2 [,] ( |_ `  A ) )
3312a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  0  e.  RR )
34 id 20 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR )
35 2re 10069 . . . . . . . 8  |-  2  e.  RR
36 2pos 10082 . . . . . . . 8  |-  0  <  2
3712, 35, 36ltleii 9196 . . . . . . 7  |-  0  <_  2
3837a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  0  <_  2 )
39 flle 11208 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
40 iccss 10978 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
2  /\  ( |_ `  A )  <_  A
) )  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
4133, 34, 38, 39, 40syl22anc 1185 . . . . 5  |-  ( A  e.  RR  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
4232, 41syl5ss 3359 . . . 4  |-  ( A  e.  RR  ->  (
( 2 [,] ( |_ `  A ) )  i^i  ZZ )  C_  ( 0 [,] A
) )
4331, 42eqsstrd 3382 . . 3  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  C_  ( 0 [,] A
) )
44 ssrin 3566 . . 3  |-  ( ( 2 ... ( |_
`  A ) ) 
C_  ( 0 [,] A )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4543, 44syl 16 . 2  |-  ( A  e.  RR  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4628, 45eqssd 3365 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990    <_ cle 9121   2c2 10049   ZZcz 10282   ZZ>=cuz 10488   [,]cicc 10919   ...cfz 11043   |_cfl 11201   Primecprime 13079
This theorem is referenced by:  ppisval2  20887  ppifi  20888  ppival2  20911  chtfl  20932  chtprm  20936  chtnprm  20937  ppifl  20943  cht1  20948  chtlepsi  20990  chpval2  21002  chpub  21004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-icc 10923  df-fz 11044  df-fl 11202  df-dvds 12853  df-prm 13080
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