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Theorem ppisval 20394
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 3424 . . . . . . . 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 3212 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  Prime )
4 prmuz2 12823 . . . . . . 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 12809 . . . . . . . 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 10974 . . . . . . . 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 3423 . . . . . . . . . . 11  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1110, 2sseldi 3212 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 0 [,] A ) )
12 0re 8883 . . . . . . . . . . 11  |-  0  e.  RR
13 simpl 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
14 elicc2 10762 . . . . . . . . . . 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 10984 . . . . . . . . 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 10283 . . . . . . 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 10839 . . . . . 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 3392 . . . . 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 3219 . 2  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
29 2z 10101 . . . . 5  |-  2  e.  ZZ
30 fzval2 10832 . . . . 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 3423 . . . . 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 9860 . . . . . . . 8  |-  2  e.  RR
36 2pos 9873 . . . . . . . 8  |-  0  <  2
3712, 35, 36ltleii 8986 . . . . . . 7  |-  0  <_  2
3837a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  0  <_  2 )
39 flle 10978 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
40 iccss 10765 . . . . . 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 3224 . . . 4  |-  ( A  e.  RR  ->  (
( 2 [,] ( |_ `  A ) )  i^i  ZZ )  C_  ( 0 [,] A
) )
4331, 42eqsstrd 3246 . . 3  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  C_  ( 0 [,] A
) )
44 ssrin 3428 . . 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 3230 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 1633    e. wcel 1701    i^i cin 3185    C_ wss 3186   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   RRcr 8781   0cc0 8782    <_ cle 8913   2c2 9840   ZZcz 10071   ZZ>=cuz 10277   [,]cicc 10706   ...cfz 10829   |_cfl 10971   Primecprime 12805
This theorem is referenced by:  ppisval2  20395  ppifi  20396  ppival2  20419  chtfl  20440  chtprm  20444  chtnprm  20445  ppifl  20451  cht1  20456  chtlepsi  20498  chpval2  20510  chpub  20512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-icc 10710  df-fz 10830  df-fl 10972  df-dvds 12579  df-prm 12806
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