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Theorem ppiwordi 20400
Description: The prime pi function is weakly increasing. (Contributed by Mario Carneiro, 19-Sep-2014.)
Assertion
Ref Expression
ppiwordi  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π `  B ) )

Proof of Theorem ppiwordi
StepHypRef Expression
1 simp2 956 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR )
2 ppifi 20343 . . . . 5  |-  ( B  e.  RR  ->  (
( 0 [,] B
)  i^i  Prime )  e. 
Fin )
31, 2syl 15 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] B
)  i^i  Prime )  e. 
Fin )
4 0re 8838 . . . . . . 7  |-  0  e.  RR
54a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  RR )
6 0le0 9827 . . . . . . 7  |-  0  <_  0
76a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  <_  0 )
8 simp3 957 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
9 iccss 10718 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  B  e.  RR )  /\  ( 0  <_ 
0  /\  A  <_  B ) )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
105, 1, 7, 8, 9syl22anc 1183 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
11 ssrin 3394 . . . . 5  |-  ( ( 0 [,] A ) 
C_  ( 0 [,] B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
1210, 11syl 15 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
13 ssdomg 6907 . . . 4  |-  ( ( ( 0 [,] B
)  i^i  Prime )  e. 
Fin  ->  ( ( ( 0 [,] A )  i^i  Prime )  C_  (
( 0 [,] B
)  i^i  Prime )  -> 
( ( 0 [,] A )  i^i  Prime )  ~<_  ( ( 0 [,] B )  i^i  Prime ) ) )
143, 12, 13sylc 56 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  ~<_  ( ( 0 [,] B
)  i^i  Prime ) )
15 ppifi 20343 . . . . 5  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
16153ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
17 hashdom 11361 . . . 4  |-  ( ( ( ( 0 [,] A )  i^i  Prime )  e.  Fin  /\  (
( 0 [,] B
)  i^i  Prime )  e. 
Fin )  ->  (
( # `  ( ( 0 [,] A )  i^i  Prime ) )  <_ 
( # `  ( ( 0 [,] B )  i^i  Prime ) )  <->  ( (
0 [,] A )  i^i  Prime )  ~<_  ( ( 0 [,] B )  i^i  Prime ) ) )
1816, 3, 17syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( # `  ( ( 0 [,] A )  i^i  Prime ) )  <_ 
( # `  ( ( 0 [,] B )  i^i  Prime ) )  <->  ( (
0 [,] A )  i^i  Prime )  ~<_  ( ( 0 [,] B )  i^i  Prime ) ) )
1914, 18mpbird 223 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( # `
 ( ( 0 [,] A )  i^i 
Prime ) )  <_  ( # `
 ( ( 0 [,] B )  i^i 
Prime ) ) )
20 ppival 20365 . . 3  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
21203ad2ant1 976 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
22 ppival 20365 . . 3  |-  ( B  e.  RR  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
231, 22syl 15 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
2419, 21, 233brtr4d 4053 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ 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    ~<_ cdom 6861   Fincfn 6863   RRcr 8736   0cc0 8737    <_ cle 8868   [,]cicc 10659   #chash 11337   Primecprime 12758  πcppi 20331
This theorem is referenced by:  ppinncl  20412  ppieq0  20414  ppiub  20443  chebbnd1lem1  20618  chebbnd1lem3  20620
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-card 7572  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-hash 11338  df-dvds 12532  df-prm 12759  df-ppi 20337
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