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Theorem resqrexlemsqa 11065
Description: Lemma for resqrex 11067. The square of a limit is  A. (Contributed by Jim Kingdon, 7-Aug-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 )
resqrexlemgt0.rr  |-  ( ph  ->  L  e.  RR )
resqrexlemgt0.lim  |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) ) )
Assertion
Ref Expression
resqrexlemsqa  |-  ( ph  ->  ( L ^ 2 )  =  A )
Distinct variable groups:    A, e, j   
y, A, z    e, F, j    y, F, z   
i, F    e, L, j, i    y, L, z   
e, i, j    ph, y,
z
Allowed substitution hints:    ph( e, i, j)    A( i)

Proof of Theorem resqrexlemsqa
Dummy variables  a  b  c  d  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resqrexlemex.seq . . . . . . 7  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) )
2 resqrexlemex.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
3 resqrexlemex.agt0 . . . . . . 7  |-  ( ph  ->  0  <_  A )
41, 2, 3resqrexlemf 11048 . . . . . 6  |-  ( ph  ->  F : NN --> RR+ )
54ffvelcdmda 5672 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  RR+ )
6 2z 9311 . . . . . 6  |-  2  e.  ZZ
76a1i 9 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  2  e.  ZZ )
85, 7rpexpcld 10709 . . . 4  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( F `  x ) ^ 2 )  e.  RR+ )
9 eqid 2189 . . . 4  |-  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) )  =  ( x  e.  NN  |->  ( ( F `
 x ) ^
2 ) )
108, 9fmptd 5691 . . 3  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR+ )
11 rpssre 9694 . . . 4  |-  RR+  C_  RR
1211a1i 9 . . 3  |-  ( ph  -> 
RR+  C_  RR )
1310, 12fssd 5397 . 2  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR )
14 resqrexlemgt0.rr . . 3  |-  ( ph  ->  L  e.  RR )
1514resqcld 10711 . 2  |-  ( ph  ->  ( L ^ 2 )  e.  RR )
16 resqrexlemgt0.lim . . . 4  |-  ( ph  ->  A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) ) )
17 oveq2 5904 . . . . . . . . 9  |-  ( e  =  a  ->  ( L  +  e )  =  ( L  +  a ) )
1817breq2d 4030 . . . . . . . 8  |-  ( e  =  a  ->  (
( F `  i
)  <  ( L  +  e )  <->  ( F `  i )  <  ( L  +  a )
) )
19 oveq2 5904 . . . . . . . . 9  |-  ( e  =  a  ->  (
( F `  i
)  +  e )  =  ( ( F `
 i )  +  a ) )
2019breq2d 4030 . . . . . . . 8  |-  ( e  =  a  ->  ( L  <  ( ( F `
 i )  +  e )  <->  L  <  ( ( F `  i
)  +  a ) ) )
2118, 20anbi12d 473 . . . . . . 7  |-  ( e  =  a  ->  (
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <-> 
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) ) )
2221rexralbidv 2516 . . . . . 6  |-  ( e  =  a  ->  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <->  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) ) )
2322cbvralv 2718 . . . . 5  |-  ( A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <->  A. a  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) )
24 fveq2 5534 . . . . . . . 8  |-  ( j  =  b  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  b )
)
2524raleqdv 2692 . . . . . . 7  |-  ( j  =  b  ->  ( A. i  e.  ( ZZ>=
`  j ) ( ( F `  i
)  <  ( L  +  a )  /\  L  <  ( ( F `
 i )  +  a ) )  <->  A. i  e.  ( ZZ>= `  b )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) ) )
2625cbvrexv 2719 . . . . . 6  |-  ( E. j  e.  NN  A. i  e.  ( ZZ>= `  j ) ( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <->  E. b  e.  NN  A. i  e.  ( ZZ>= `  b )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) )
2726ralbii 2496 . . . . 5  |-  ( A. a  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <->  A. a  e.  RR+  E. b  e.  NN  A. i  e.  ( ZZ>= `  b )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) )
28 fveq2 5534 . . . . . . . . . 10  |-  ( i  =  c  ->  ( F `  i )  =  ( F `  c ) )
2928breq1d 4028 . . . . . . . . 9  |-  ( i  =  c  ->  (
( F `  i
)  <  ( L  +  a )  <->  ( F `  c )  <  ( L  +  a )
) )
3028oveq1d 5911 . . . . . . . . . 10  |-  ( i  =  c  ->  (
( F `  i
)  +  a )  =  ( ( F `
 c )  +  a ) )
3130breq2d 4030 . . . . . . . . 9  |-  ( i  =  c  ->  ( L  <  ( ( F `
 i )  +  a )  <->  L  <  ( ( F `  c
)  +  a ) ) )
3229, 31anbi12d 473 . . . . . . . 8  |-  ( i  =  c  ->  (
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <-> 
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) ) )
3332cbvralv 2718 . . . . . . 7  |-  ( A. i  e.  ( ZZ>= `  b ) ( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <->  A. c  e.  ( ZZ>= `  b )
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) )
3433rexbii 2497 . . . . . 6  |-  ( E. b  e.  NN  A. i  e.  ( ZZ>= `  b ) ( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <->  E. b  e.  NN  A. c  e.  ( ZZ>= `  b )
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) )
3534ralbii 2496 . . . . 5  |-  ( A. a  e.  RR+  E. b  e.  NN  A. i  e.  ( ZZ>= `  b )
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <->  A. a  e.  RR+  E. b  e.  NN  A. c  e.  ( ZZ>= `  b )
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) )
3623, 27, 353bitri 206 . . . 4  |-  ( A. e  e.  RR+  E. j  e.  NN  A. i  e.  ( ZZ>= `  j )
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <->  A. a  e.  RR+  E. b  e.  NN  A. c  e.  ( ZZ>= `  b )
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) )
3716, 36sylib 122 . . 3  |-  ( ph  ->  A. a  e.  RR+  E. b  e.  NN  A. c  e.  ( ZZ>= `  b ) ( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) )
381, 2, 3, 14, 37, 9resqrexlemglsq 11063 . 2  |-  ( ph  ->  A. a  e.  RR+  E. b  e.  NN  A. d  e.  ( ZZ>= `  b ) ( ( ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) `  d
)  <  ( ( L ^ 2 )  +  a )  /\  ( L ^ 2 )  < 
( ( ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) `
 d )  +  a ) ) )
391, 2, 3, 14, 37, 9resqrexlemga 11064 . 2  |-  ( ph  ->  A. a  e.  RR+  E. b  e.  NN  A. d  e.  ( ZZ>= `  b ) ( ( ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) `  d
)  <  ( A  +  a )  /\  A  <  ( ( ( x  e.  NN  |->  ( ( F `  x
) ^ 2 ) ) `  d )  +  a ) ) )
4013, 15, 38, 2, 39recvguniq 11036 1  |-  ( ph  ->  ( L ^ 2 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469    C_ wss 3144   {csn 3607   class class class wbr 4018    |-> cmpt 4079    X. cxp 4642   ` cfv 5235  (class class class)co 5896    e. cmpo 5898   RRcr 7840   0cc0 7841   1c1 7842    + caddc 7844    < clt 8022    <_ cle 8023    / cdiv 8659   NNcn 8949   2c2 9000   ZZcz 9283   ZZ>=cuz 9558   RR+crp 9683    seqcseq 10476   ^cexp 10550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-rp 9684  df-seqfrec 10477  df-exp 10551
This theorem is referenced by:  resqrexlemex  11066
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