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Theorem ppisval 20839
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 3522 . . . . . . . 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 3306 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  Prime )
4 prmuz2 13052 . . . . . . 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 13038 . . . . . . . 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 11159 . . . . . . . 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 3521 . . . . . . . . . . 11  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1110, 2sseldi 3306 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 0 [,] A ) )
12 0re 9047 . . . . . . . . . . 11  |-  0  e.  RR
13 simpl 444 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
14 elicc2 10931 . . . . . . . . . . 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 11169 . . . . . . . . 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 10450 . . . . . . 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 11009 . . . . . 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 3490 . . . . 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 3314 . 2  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 2z 10268 . . . . 5  |-  2  e.  ZZ
30 fzval2 11002 . . . . 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 3521 . . . . 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 10025 . . . . . . . 8  |-  2  e.  RR
36 2pos 10038 . . . . . . . 8  |-  0  <  2
3712, 35, 36ltleii 9152 . . . . . . 7  |-  0  <_  2
3837a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  0  <_  2 )
39 flle 11163 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
40 iccss 10934 . . . . . 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 3319 . . . 4  |-  ( A  e.  RR  ->  (
( 2 [,] ( |_ `  A ) )  i^i  ZZ )  C_  ( 0 [,] A
) )
4331, 42eqsstrd 3342 . . 3  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  C_  ( 0 [,] A
) )
44 ssrin 3526 . . 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 3325 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 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    <_ cle 9077   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   [,]cicc 10875   ...cfz 10999   |_cfl 11156   Primecprime 13034
This theorem is referenced by:  ppisval2  20840  ppifi  20841  ppival2  20864  chtfl  20885  chtprm  20889  chtnprm  20890  ppifl  20896  cht1  20901  chtlepsi  20943  chpval2  20955  chpub  20957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-icc 10879  df-fz 11000  df-fl 11157  df-dvds 12808  df-prm 13035
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