MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ppiwordi Structured version   Unicode version

Theorem ppiwordi 22512
Description: The prime-counting function π 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 989 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR )
2 ppifi 22455 . . . . 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 0red 9399 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  RR )
5 0le0 10423 . . . . . . 7  |-  0  <_  0
65a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  <_  0 )
7 simp3 990 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
8 iccss 11375 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  B  e.  RR )  /\  ( 0  <_ 
0  /\  A  <_  B ) )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
94, 1, 6, 7, 8syl22anc 1219 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
10 ssrin 3587 . . . . 5  |-  ( ( 0 [,] A ) 
C_  ( 0 [,] B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
119, 10syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  C_  ( ( 0 [,] B )  i^i  Prime ) )
12 ssdomg 7367 . . . 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 ) ) )
133, 11, 12sylc 60 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  ~<_  ( ( 0 [,] B
)  i^i  Prime ) )
14 ppifi 22455 . . . . 5  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
15143ad2ant1 1009 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
16 hashdom 12154 . . . 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 ) ) )
1715, 3, 16syl2anc 661 . . 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 ) ) )
1813, 17mpbird 232 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( # `
 ( ( 0 [,] A )  i^i 
Prime ) )  <_  ( # `
 ( ( 0 [,] B )  i^i 
Prime ) ) )
19 ppival 22477 . . 3  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
20193ad2ant1 1009 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
21 ppival 22477 . . 3  |-  ( B  e.  RR  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
221, 21syl 16 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  B )  =  (
# `  ( (
0 [,] B )  i^i  Prime ) ) )
2318, 20, 223brtr4d 4334 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  <_  (π `  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3339    C_ wss 3340   class class class wbr 4304   ` cfv 5430  (class class class)co 6103    ~<_ cdom 7320   Fincfn 7322   RRcr 9293   0cc0 9294    <_ cle 9431   [,]cicc 11315   #chash 12115   Primecprime 13775  πcppi 22443
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-icc 11319  df-fz 11450  df-fl 11654  df-hash 12116  df-dvds 13548  df-prm 13776  df-ppi 22449
This theorem is referenced by:  ppinncl  22524  ppieq0  22526  ppiub  22555  chebbnd1lem1  22730  chebbnd1lem3  22732
  Copyright terms: Public domain W3C validator