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Theorem ppiwordi 23562
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 997 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  B  e.  RR )
2 ppifi 23505 . . . . 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 9614 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  e.  RR )
5 0le0 10646 . . . . . . 7  |-  0  <_  0
65a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  0  <_  0 )
7 simp3 998 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
8 iccss 11617 . . . . . 6  |-  ( ( ( 0  e.  RR  /\  B  e.  RR )  /\  ( 0  <_ 
0  /\  A  <_  B ) )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
94, 1, 6, 7, 8syl22anc 1229 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
0 [,] A ) 
C_  ( 0 [,] B ) )
10 ssrin 3719 . . . . 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 7580 . . . 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 23505 . . . . 5  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
15143ad2ant1 1017 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
16 hashdom 12450 . . . 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 23527 . . 3  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
20193ad2ant1 1017 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
21 ppival 23527 . . 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 4486 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 973    = wceq 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296    ~<_ cdom 7533   Fincfn 7535   RRcr 9508   0cc0 9509    <_ cle 9646   [,]cicc 11557   #chash 12408   Primecprime 14229  πcppi 23493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-icc 11561  df-fz 11698  df-fl 11932  df-hash 12409  df-dvds 13999  df-prm 14230  df-ppi 23499
This theorem is referenced by:  ppinncl  23574  ppieq0  23576  ppiub  23605  chebbnd1lem1  23780  chebbnd1lem3  23782
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