ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  resqrexlemfp1 Unicode version

Theorem resqrexlemfp1 10749
Description: Lemma for resqrex 10766. Recursion rule. This sequence is the ancient method for computing square roots, often known as the babylonian method, although known to many ancient cultures. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.)
Hypotheses
Ref Expression
resqrexlemex.seq  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) )
resqrexlemex.a  |-  ( ph  ->  A  e.  RR )
resqrexlemex.agt0  |-  ( ph  ->  0  <_  A )
Assertion
Ref Expression
resqrexlemfp1  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( N  + 
1 ) )  =  ( ( ( F `
 N )  +  ( A  /  ( F `  N )
) )  /  2
) )
Distinct variable groups:    y, A, z    ph, y, z
Allowed substitution hints:    F( y, z)    N( y, z)

Proof of Theorem resqrexlemfp1
Dummy variables  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnnuz 9330 . . . . . 6  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
21biimpi 119 . . . . 5  |-  ( N  e.  NN  ->  N  e.  ( ZZ>= `  1 )
)
32adantl 275 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  N  e.  ( ZZ>= `  1 )
)
4 elnnuz 9330 . . . . . 6  |-  ( a  e.  NN  <->  a  e.  ( ZZ>= `  1 )
)
5 resqrexlemex.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
6 resqrexlemex.agt0 . . . . . . 7  |-  ( ph  ->  0  <_  A )
75, 6resqrexlem1arp 10745 . . . . . 6  |-  ( (
ph  /\  a  e.  NN )  ->  ( ( NN  X.  { ( 1  +  A ) } ) `  a
)  e.  RR+ )
84, 7sylan2br 286 . . . . 5  |-  ( (
ph  /\  a  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { ( 1  +  A ) } ) `  a )  e.  RR+ )
98adantlr 468 . . . 4  |-  ( ( ( ph  /\  N  e.  NN )  /\  a  e.  ( ZZ>= `  1 )
)  ->  ( ( NN  X.  { ( 1  +  A ) } ) `  a )  e.  RR+ )
105, 6resqrexlemp1rp 10746 . . . . 5  |-  ( (
ph  /\  ( a  e.  RR+  /\  b  e.  RR+ ) )  ->  (
a ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) b )  e.  RR+ )
1110adantlr 468 . . . 4  |-  ( ( ( ph  /\  N  e.  NN )  /\  (
a  e.  RR+  /\  b  e.  RR+ ) )  -> 
( a ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) ) b )  e.  RR+ )
123, 9, 11seq3p1 10203 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  (  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) ) ,  ( NN  X.  {
( 1  +  A
) } ) ) `
 ( N  + 
1 ) )  =  ( (  seq 1
( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) ,  ( NN  X.  { ( 1  +  A ) } ) ) `  N ) ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) ) ( ( NN  X.  {
( 1  +  A
) } ) `  ( N  +  1
) ) ) )
13 resqrexlemex.seq . . . 4  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) )
1413fveq1i 5390 . . 3  |-  ( F `
 ( N  + 
1 ) )  =  (  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ) `  ( N  +  1 ) )
1513fveq1i 5390 . . . 4  |-  ( F `
 N )  =  (  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ) `  N )
1615oveq1i 5752 . . 3  |-  ( ( F `  N ) ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ( ( NN  X.  { ( 1  +  A ) } ) `
 ( N  + 
1 ) ) )  =  ( (  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) ) ,  ( NN  X.  {
( 1  +  A
) } ) ) `
 N ) ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ( ( NN  X.  { ( 1  +  A ) } ) `
 ( N  + 
1 ) ) )
1712, 14, 163eqtr4g 2175 . 2  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( N  + 
1 ) )  =  ( ( F `  N ) ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) ) ( ( NN  X.  {
( 1  +  A
) } ) `  ( N  +  1
) ) ) )
18 id 19 . . . . . . 7  |-  ( y  =  c  ->  y  =  c )
19 oveq2 5750 . . . . . . 7  |-  ( y  =  c  ->  ( A  /  y )  =  ( A  /  c
) )
2018, 19oveq12d 5760 . . . . . 6  |-  ( y  =  c  ->  (
y  +  ( A  /  y ) )  =  ( c  +  ( A  /  c
) ) )
2120oveq1d 5757 . . . . 5  |-  ( y  =  c  ->  (
( y  +  ( A  /  y ) )  /  2 )  =  ( ( c  +  ( A  / 
c ) )  / 
2 ) )
22 eqidd 2118 . . . . 5  |-  ( z  =  d  ->  (
( c  +  ( A  /  c ) )  /  2 )  =  ( ( c  +  ( A  / 
c ) )  / 
2 ) )
2321, 22cbvmpov 5819 . . . 4  |-  ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) )  =  ( c  e.  RR+ ,  d  e.  RR+  |->  ( ( c  +  ( A  /  c ) )  /  2 ) )
2423a1i 9 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  / 
y ) )  / 
2 ) )  =  ( c  e.  RR+ ,  d  e.  RR+  |->  ( ( c  +  ( A  /  c ) )  /  2 ) ) )
25 id 19 . . . . . 6  |-  ( c  =  ( F `  N )  ->  c  =  ( F `  N ) )
26 oveq2 5750 . . . . . 6  |-  ( c  =  ( F `  N )  ->  ( A  /  c )  =  ( A  /  ( F `  N )
) )
2725, 26oveq12d 5760 . . . . 5  |-  ( c  =  ( F `  N )  ->  (
c  +  ( A  /  c ) )  =  ( ( F `
 N )  +  ( A  /  ( F `  N )
) ) )
2827oveq1d 5757 . . . 4  |-  ( c  =  ( F `  N )  ->  (
( c  +  ( A  /  c ) )  /  2 )  =  ( ( ( F `  N )  +  ( A  / 
( F `  N
) ) )  / 
2 ) )
2928ad2antrl 481 . . 3  |-  ( ( ( ph  /\  N  e.  NN )  /\  (
c  =  ( F `
 N )  /\  d  =  ( ( NN  X.  { ( 1  +  A ) } ) `  ( N  +  1 ) ) ) )  ->  (
( c  +  ( A  /  c ) )  /  2 )  =  ( ( ( F `  N )  +  ( A  / 
( F `  N
) ) )  / 
2 ) )
3013, 5, 6resqrexlemf 10747 . . . 4  |-  ( ph  ->  F : NN --> RR+ )
3130ffvelrnda 5523 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 N )  e.  RR+ )
32 peano2nn 8700 . . . 4  |-  ( N  e.  NN  ->  ( N  +  1 )  e.  NN )
335, 6resqrexlem1arp 10745 . . . 4  |-  ( (
ph  /\  ( N  +  1 )  e.  NN )  ->  (
( NN  X.  {
( 1  +  A
) } ) `  ( N  +  1
) )  e.  RR+ )
3432, 33sylan2 284 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( NN  X.  { ( 1  +  A ) } ) `  ( N  +  1 ) )  e.  RR+ )
3531rpred 9451 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 N )  e.  RR )
365adantr 274 . . . . . 6  |-  ( (
ph  /\  N  e.  NN )  ->  A  e.  RR )
3736, 31rerpdivcld 9483 . . . . 5  |-  ( (
ph  /\  N  e.  NN )  ->  ( A  /  ( F `  N ) )  e.  RR )
3835, 37readdcld 7763 . . . 4  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( F `  N )  +  ( A  / 
( F `  N
) ) )  e.  RR )
3938rehalfcld 8934 . . 3  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( ( F `  N
)  +  ( A  /  ( F `  N ) ) )  /  2 )  e.  RR )
4024, 29, 31, 34, 39ovmpod 5866 . 2  |-  ( (
ph  /\  N  e.  NN )  ->  ( ( F `  N ) ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ( ( NN  X.  { ( 1  +  A ) } ) `
 ( N  + 
1 ) ) )  =  ( ( ( F `  N )  +  ( A  / 
( F `  N
) ) )  / 
2 ) )
4117, 40eqtrd 2150 1  |-  ( (
ph  /\  N  e.  NN )  ->  ( F `
 ( N  + 
1 ) )  =  ( ( ( F `
 N )  +  ( A  /  ( F `  N )
) )  /  2
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   {csn 3497   class class class wbr 3899    X. cxp 4507   ` cfv 5093  (class class class)co 5742    e. cmpo 5744   RRcr 7587   0cc0 7588   1c1 7589    + caddc 7591    <_ cle 7769    / cdiv 8400   NNcn 8688   2c2 8739   ZZ>=cuz 9294   RR+crp 9409    seqcseq 10186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-uz 9295  df-rp 9410  df-seqfrec 10187
This theorem is referenced by:  resqrexlemover  10750  resqrexlemdec  10751  resqrexlemlo  10753  resqrexlemcalc1  10754
  Copyright terms: Public domain W3C validator