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Theorem ppisval 22440
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 3570 . . . . . . . 8  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
2 simpr 461 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
0 [,] A )  i^i  Prime ) )
31, 2sseldi 3353 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  Prime )
4 prmuz2 13780 . . . . . . 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 13766 . . . . . . . 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 11644 . . . . . . . 8  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  ZZ )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ZZ )
10 inss1 3569 . . . . . . . . . . 11  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1110, 2sseldi 3353 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 0 [,] A ) )
12 0re 9385 . . . . . . . . . . 11  |-  0  e.  RR
13 simpl 457 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  A  e.  RR )
14 elicc2 11359 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1512, 13, 14sylancr 663 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  ( 0 [,] A )  <-> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) ) )
1611, 15mpbid 210 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  e.  RR  /\  0  <_  x  /\  x  <_  A ) )
1716simp3d 1002 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  A )
18 flge 11654 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  x  e.  ZZ )  ->  ( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
197, 18syldan 470 . . . . . . . 8  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( x  <_  A  <->  x  <_  ( |_ `  A ) ) )
2017, 19mpbid 210 . . . . . . 7  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  <_  ( |_ `  A ) )
21 eluz2 10866 . . . . . . 7  |-  ( ( |_ `  A )  e.  ( ZZ>= `  x
)  <->  ( x  e.  ZZ  /\  ( |_
`  A )  e.  ZZ  /\  x  <_ 
( |_ `  A
) ) )
227, 9, 20, 21syl3anbrc 1172 . . . . . 6  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  -> 
( |_ `  A
)  e.  ( ZZ>= `  x ) )
23 elfzuzb 11446 . . . . . 6  |-  ( x  e.  ( 2 ... ( |_ `  A
) )  <->  ( x  e.  ( ZZ>= `  2 )  /\  ( |_ `  A
)  e.  ( ZZ>= `  x ) ) )
245, 22, 23sylanbrc 664 . . . . 5  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( 2 ... ( |_ `  A ) ) )
2524, 3elind 3539 . . . 4  |-  ( ( A  e.  RR  /\  x  e.  ( (
0 [,] A )  i^i  Prime ) )  ->  x  e.  ( (
2 ... ( |_ `  A ) )  i^i 
Prime ) )
2625ex 434 . . 3  |-  ( A  e.  RR  ->  (
x  e.  ( ( 0 [,] A )  i^i  Prime )  ->  x  e.  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) ) )
2726ssrdv 3361 . 2  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
28 2z 10677 . . . . 5  |-  2  e.  ZZ
29 fzval2 11439 . . . . 5  |-  ( ( 2  e.  ZZ  /\  ( |_ `  A )  e.  ZZ )  -> 
( 2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A ) )  i^i 
ZZ ) )
3028, 8, 29sylancr 663 . . . 4  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  =  ( ( 2 [,] ( |_ `  A
) )  i^i  ZZ ) )
31 inss1 3569 . . . . 5  |-  ( ( 2 [,] ( |_
`  A ) )  i^i  ZZ )  C_  ( 2 [,] ( |_ `  A ) )
3212a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  0  e.  RR )
33 id 22 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR )
34 0le2 10411 . . . . . . 7  |-  0  <_  2
3534a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  0  <_  2 )
36 flle 11648 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  A )  <_  A )
37 iccss 11362 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( 0  <_ 
2  /\  ( |_ `  A )  <_  A
) )  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
3832, 33, 35, 36, 37syl22anc 1219 . . . . 5  |-  ( A  e.  RR  ->  (
2 [,] ( |_
`  A ) ) 
C_  ( 0 [,] A ) )
3931, 38syl5ss 3366 . . . 4  |-  ( A  e.  RR  ->  (
( 2 [,] ( |_ `  A ) )  i^i  ZZ )  C_  ( 0 [,] A
) )
4030, 39eqsstrd 3389 . . 3  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  A ) )  C_  ( 0 [,] A
) )
41 ssrin 3574 . . 3  |-  ( ( 2 ... ( |_
`  A ) ) 
C_  ( 0 [,] A )  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4240, 41syl 16 . 2  |-  ( A  e.  RR  ->  (
( 2 ... ( |_ `  A ) )  i^i  Prime )  C_  (
( 0 [,] A
)  i^i  Prime ) )
4327, 42eqssd 3372 1  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3326    C_ wss 3327   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   RRcr 9280   0cc0 9281    <_ cle 9418   2c2 10370   ZZcz 10645   ZZ>=cuz 10860   [,]cicc 11302   ...cfz 11436   |_cfl 11639   Primecprime 13762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-2o 6920  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-n0 10579  df-z 10646  df-uz 10861  df-icc 11306  df-fz 11437  df-fl 11641  df-dvds 13535  df-prm 13763
This theorem is referenced by:  ppisval2  22441  ppifi  22442  ppival2  22465  chtfl  22486  chtprm  22490  chtnprm  22491  ppifl  22497  cht1  22502  chtlepsi  22544  chpval2  22556  chpub  22558
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