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Theorem ruclem10 13632
Description: Lemma for ruc 13636. Every first component of the  G sequence is less than every second component. That is, the sequences form a chain a1  < a2 
<...  < b2  < b1, where ai are the first components and bi are the second components. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1  |-  ( ph  ->  F : NN --> RR )
ruc.2  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
ruc.4  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
ruc.5  |-  G  =  seq 0 ( D ,  C )
ruclem10.6  |-  ( ph  ->  M  e.  NN0 )
ruclem10.7  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
ruclem10  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Distinct variable groups:    x, m, y, F    m, G, x, y    m, M, x, y    m, N, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem10
StepHypRef Expression
1 ruc.1 . . . . 5  |-  ( ph  ->  F : NN --> RR )
2 ruc.2 . . . . 5  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
3 ruc.4 . . . . 5  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
4 ruc.5 . . . . 5  |-  G  =  seq 0 ( D ,  C )
51, 2, 3, 4ruclem6 13628 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 ruclem10.6 . . . 4  |-  ( ph  ->  M  e.  NN0 )
75, 6ffvelrnd 5946 . . 3  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
8 xp1st 6709 . . 3  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
97, 8syl 16 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
10 ruclem10.7 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
11 ifcl 3932 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
1210, 6, 11syl2anc 661 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
135, 12ffvelrnd 5946 . . 3  |-  ( ph  ->  ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR ) )
14 xp1st 6709 . . 3  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
1513, 14syl 16 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
165, 10ffvelrnd 5946 . . 3  |-  ( ph  ->  ( G `  N
)  e.  ( RR 
X.  RR ) )
17 xp2nd 6710 . . 3  |-  ( ( G `  N )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  N
) )  e.  RR )
1816, 17syl 16 . 2  |-  ( ph  ->  ( 2nd `  ( G `  N )
)  e.  RR )
196nn0red 10741 . . . . . 6  |-  ( ph  ->  M  e.  RR )
2010nn0red 10741 . . . . . 6  |-  ( ph  ->  N  e.  RR )
21 max1 11261 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
2219, 20, 21syl2anc 661 . . . . 5  |-  ( ph  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
236nn0zd 10849 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2412nn0zd 10849 . . . . . 6  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  ZZ )
25 eluz 10978 . . . . . 6  |-  ( ( M  e.  ZZ  /\  if ( M  <_  N ,  N ,  M )  e.  ZZ )  -> 
( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
2623, 24, 25syl2anc 661 . . . . 5  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
2722, 26mpbird 232 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M ) )
281, 2, 3, 4, 6, 27ruclem9 13631 . . 3  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
2928simpld 459 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) ) )
30 xp2nd 6710 . . . 4  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
3113, 30syl 16 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
321, 2, 3, 4ruclem8 13630 . . . 4  |-  ( (
ph  /\  if ( M  <_  N ,  N ,  M )  e.  NN0 )  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
3312, 32mpdan 668 . . 3  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
34 max2 11263 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3519, 20, 34syl2anc 661 . . . . . 6  |-  ( ph  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3610nn0zd 10849 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
37 eluz 10978 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  if ( M  <_  N ,  N ,  M )  e.  ZZ )  -> 
( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
3836, 24, 37syl2anc 661 . . . . . 6  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
3935, 38mpbird 232 . . . . 5  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N ) )
401, 2, 3, 4, 10, 39ruclem9 13631 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  N )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N ) ) ) )
4140simprd 463 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N )
) )
4215, 31, 18, 33, 41ltletrd 9635 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  N )
) )
439, 15, 18, 29, 42lelttrd 9633 1  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758   [_csb 3389    u. cun 3427   ifcif 3892   {csn 3978   <.cop 3984   class class class wbr 4393    X. cxp 4939   -->wf 5515   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1stc1st 6678   2ndc2nd 6679   RRcr 9385   0cc0 9386   1c1 9387    + caddc 9389    < clt 9522    <_ cle 9523    / cdiv 10097   NNcn 10426   2c2 10475   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965    seqcseq 11916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-seq 11917
This theorem is referenced by:  ruclem11  13633  ruclem12  13634
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