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Theorem ppiwordi 20937
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 958 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR )
2 ppifi 20880 . . . . 5  |-  ( B  e.  RR  ->  (
( 0 [,] B
)  i^i  Prime )  e. 
Fin )
31, 2syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] B
)  i^i  Prime )  e. 
Fin )
4 0re 9083 . . . . . . 7  |-  0  e.  RR
54a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  RR )
6 0le0 10073 . . . . . . 7  |-  0  <_  0
76a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  <_  0 )
8 simp3 959 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
9 iccss 10970 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  B  e.  RR )  /\  ( 0  <_ 
0  /\  A  <_  B ) )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
105, 1, 7, 8, 9syl22anc 1185 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
11 ssrin 3558 . . . . 5  |-  ( ( 0 [,] A ) 
C_  ( 0 [,] B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
1210, 11syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
13 ssdomg 7145 . . . 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 58 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  ~<_  ( ( 0 [,] B
)  i^i  Prime ) )
15 ppifi 20880 . . . . 5  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
16153ad2ant1 978 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
17 hashdom 11645 . . . 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 643 . . 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 224 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( # `
 ( ( 0 [,] A )  i^i 
Prime ) )  <_  ( # `
 ( ( 0 [,] B )  i^i 
Prime ) ) )
20 ppival 20902 . . 3  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
21203ad2ant1 978 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
22 ppival 20902 . . 3  |-  ( B  e.  RR  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
231, 22syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
2419, 21, 233brtr4d 4234 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    i^i cin 3311    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    ~<_ cdom 7099   Fincfn 7101   RRcr 8981   0cc0 8982    <_ cle 9113   [,]cicc 10911   #chash 11610   Primecprime 13071  πcppi 20868
This theorem is referenced by:  ppinncl  20949  ppieq0  20951  ppiub  20980  chebbnd1lem1  21155  chebbnd1lem3  21157
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-icc 10915  df-fz 11036  df-fl 11194  df-hash 11611  df-dvds 12845  df-prm 13072  df-ppi 20874
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