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Theorem resqrexlemsqa 9622
Description: Lemma for resqrex 9624. 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 ) } ) ,  RR+ )
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 ) } ) ,  RR+ )
2 resqrexlemex.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
3 resqrexlemex.agt0 . . . . . . 7  |-  ( ph  ->  0  <_  A )
41, 2, 3resqrexlemf 9605 . . . . . 6  |-  ( ph  ->  F : NN --> RR+ )
54ffvelrnda 5302 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  RR+ )
6 2z 8273 . . . . . 6  |-  2  e.  ZZ
76a1i 9 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  2  e.  ZZ )
85, 7rpexpcld 9404 . . . 4  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( F `  x ) ^ 2 )  e.  RR+ )
9 eqid 2040 . . . 4  |-  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) )  =  ( x  e.  NN  |->  ( ( F `
 x ) ^
2 ) )
108, 9fmptd 5322 . . 3  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR+ )
11 rpssre 8593 . . . 4  |-  RR+  C_  RR
1211a1i 9 . . 3  |-  ( ph  -> 
RR+  C_  RR )
1310, 12fssd 5055 . 2  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR )
14 resqrexlemgt0.rr . . 3  |-  ( ph  ->  L  e.  RR )
1514resqcld 9406 . 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 5520 . . . . . . . . 9  |-  ( e  =  a  ->  ( L  +  e )  =  ( L  +  a ) )
1817breq2d 3776 . . . . . . . 8  |-  ( e  =  a  ->  (
( F `  i
)  <  ( L  +  e )  <->  ( F `  i )  <  ( L  +  a )
) )
19 oveq2 5520 . . . . . . . . 9  |-  ( e  =  a  ->  (
( F `  i
)  +  e )  =  ( ( F `
 i )  +  a ) )
2019breq2d 3776 . . . . . . . 8  |-  ( e  =  a  ->  ( L  <  ( ( F `
 i )  +  e )  <->  L  <  ( ( F `  i
)  +  a ) ) )
2118, 20anbi12d 442 . . . . . . 7  |-  ( e  =  a  ->  (
( ( F `  i )  <  ( L  +  e )  /\  L  <  ( ( F `  i )  +  e ) )  <-> 
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) ) ) )
2221rexralbidv 2350 . . . . . 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 2533 . . . . 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 5178 . . . . . . . 8  |-  ( j  =  b  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  b )
)
2524raleqdv 2511 . . . . . . 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 2534 . . . . . 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 2330 . . . . 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 5178 . . . . . . . . . 10  |-  ( i  =  c  ->  ( F `  i )  =  ( F `  c ) )
2928breq1d 3774 . . . . . . . . 9  |-  ( i  =  c  ->  (
( F `  i
)  <  ( L  +  a )  <->  ( F `  c )  <  ( L  +  a )
) )
3028oveq1d 5527 . . . . . . . . . 10  |-  ( i  =  c  ->  (
( F `  i
)  +  a )  =  ( ( F `
 c )  +  a ) )
3130breq2d 3776 . . . . . . . . 9  |-  ( i  =  c  ->  ( L  <  ( ( F `
 i )  +  a )  <->  L  <  ( ( F `  c
)  +  a ) ) )
3229, 31anbi12d 442 . . . . . . . 8  |-  ( i  =  c  ->  (
( ( F `  i )  <  ( L  +  a )  /\  L  <  ( ( F `  i )  +  a ) )  <-> 
( ( F `  c )  <  ( L  +  a )  /\  L  <  ( ( F `  c )  +  a ) ) ) )
3332cbvralv 2533 . . . . . . 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 2331 . . . . . 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 2330 . . . . 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 195 . . . 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 127 . . 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 9620 . 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 9621 . 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 9593 1  |-  ( ph  ->  ( L ^ 2 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307    C_ wss 2917   {csn 3375   class class class wbr 3764    |-> cmpt 3818    X. cxp 4343   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   RRcr 6888   0cc0 6889   1c1 6890    + caddc 6892    < clt 7060    <_ cle 7061    / cdiv 7651   NNcn 7914   2c2 7964   ZZcz 8245   ZZ>=cuz 8473   RR+crp 8583    seqcseq 9211   ^cexp 9254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulrcl 6983  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-precex 6994  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000  ax-pre-mulgt0 7001  ax-pre-mulext 7002  ax-arch 7003
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-reap 7566  df-ap 7573  df-div 7652  df-inn 7915  df-2 7973  df-3 7974  df-4 7975  df-n0 8182  df-z 8246  df-uz 8474  df-rp 8584  df-iseq 9212  df-iexp 9255
This theorem is referenced by:  resqrexlemex  9623
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