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Theorem resqrexlemsqa 10999
Description: Lemma for resqrex 11001. 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 10982 . . . . . 6  |-  ( ph  ->  F : NN --> RR+ )
54ffvelcdmda 5643 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  ( F `
 x )  e.  RR+ )
6 2z 9252 . . . . . 6  |-  2  e.  ZZ
76a1i 9 . . . . 5  |-  ( (
ph  /\  x  e.  NN )  ->  2  e.  ZZ )
85, 7rpexpcld 10645 . . . 4  |-  ( (
ph  /\  x  e.  NN )  ->  ( ( F `  x ) ^ 2 )  e.  RR+ )
9 eqid 2175 . . . 4  |-  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) )  =  ( x  e.  NN  |->  ( ( F `
 x ) ^
2 ) )
108, 9fmptd 5662 . . 3  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR+ )
11 rpssre 9633 . . . 4  |-  RR+  C_  RR
1211a1i 9 . . 3  |-  ( ph  -> 
RR+  C_  RR )
1310, 12fssd 5370 . 2  |-  ( ph  ->  ( x  e.  NN  |->  ( ( F `  x ) ^ 2 ) ) : NN --> RR )
14 resqrexlemgt0.rr . . 3  |-  ( ph  ->  L  e.  RR )
1514resqcld 10647 . 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 5873 . . . . . . . . 9  |-  ( e  =  a  ->  ( L  +  e )  =  ( L  +  a ) )
1817breq2d 4010 . . . . . . . 8  |-  ( e  =  a  ->  (
( F `  i
)  <  ( L  +  e )  <->  ( F `  i )  <  ( L  +  a )
) )
19 oveq2 5873 . . . . . . . . 9  |-  ( e  =  a  ->  (
( F `  i
)  +  e )  =  ( ( F `
 i )  +  a ) )
2019breq2d 4010 . . . . . . . 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 2501 . . . . . 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 2701 . . . . 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 5507 . . . . . . . 8  |-  ( j  =  b  ->  ( ZZ>=
`  j )  =  ( ZZ>= `  b )
)
2524raleqdv 2676 . . . . . . 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 2702 . . . . . 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 2481 . . . . 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 5507 . . . . . . . . . 10  |-  ( i  =  c  ->  ( F `  i )  =  ( F `  c ) )
2928breq1d 4008 . . . . . . . . 9  |-  ( i  =  c  ->  (
( F `  i
)  <  ( L  +  a )  <->  ( F `  c )  <  ( L  +  a )
) )
3028oveq1d 5880 . . . . . . . . . 10  |-  ( i  =  c  ->  (
( F `  i
)  +  a )  =  ( ( F `
 c )  +  a ) )
3130breq2d 4010 . . . . . . . . 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 2701 . . . . . . 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 2482 . . . . . 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 2481 . . . . 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 10997 . 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 10998 . 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 10970 1  |-  ( ph  ->  ( L ^ 2 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454    C_ wss 3127   {csn 3589   class class class wbr 3998    |-> cmpt 4059    X. cxp 4618   ` cfv 5208  (class class class)co 5865    e. cmpo 5867   RRcr 7785   0cc0 7786   1c1 7787    + caddc 7789    < clt 7966    <_ cle 7967    / cdiv 8601   NNcn 8890   2c2 8941   ZZcz 9224   ZZ>=cuz 9499   RR+crp 9622    seqcseq 10413   ^cexp 10487
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-rp 9623  df-seqfrec 10414  df-exp 10488
This theorem is referenced by:  resqrexlemex  11000
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