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Theorem jm3.1lem2 27111
Description: Lemma for jm3.1 27113. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Hypotheses
Ref Expression
jm3.1.a  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
jm3.1.b  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
jm3.1.c  |-  ( ph  ->  N  e.  NN )
jm3.1.d  |-  ( ph  ->  ( K Yrm  ( N  + 
1 ) )  <_  A )
Assertion
Ref Expression
jm3.1lem2  |-  ( ph  ->  ( K ^ N
)  <  ( (
( ( 2  x.  A )  x.  K
)  -  ( K ^ 2 ) )  -  1 ) )

Proof of Theorem jm3.1lem2
StepHypRef Expression
1 jm3.1.b . . . 4  |-  ( ph  ->  K  e.  ( ZZ>= ` 
2 ) )
2 eluzelre 10239 . . . 4  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  RR )
31, 2syl 15 . . 3  |-  ( ph  ->  K  e.  RR )
4 jm3.1.c . . . 4  |-  ( ph  ->  N  e.  NN )
54nnnn0d 10018 . . 3  |-  ( ph  ->  N  e.  NN0 )
63, 5reexpcld 11262 . 2  |-  ( ph  ->  ( K ^ N
)  e.  RR )
7 jm3.1.a . . 3  |-  ( ph  ->  A  e.  ( ZZ>= ` 
2 ) )
8 eluzelre 10239 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
97, 8syl 15 . 2  |-  ( ph  ->  A  e.  RR )
10 2re 9815 . . . . . 6  |-  2  e.  RR
11 remulcl 8822 . . . . . 6  |-  ( ( 2  e.  RR  /\  A  e.  RR )  ->  ( 2  x.  A
)  e.  RR )
1210, 9, 11sylancr 644 . . . . 5  |-  ( ph  ->  ( 2  x.  A
)  e.  RR )
1312, 3remulcld 8863 . . . 4  |-  ( ph  ->  ( ( 2  x.  A )  x.  K
)  e.  RR )
143resqcld 11271 . . . 4  |-  ( ph  ->  ( K ^ 2 )  e.  RR )
1513, 14resubcld 9211 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  e.  RR )
16 1re 8837 . . 3  |-  1  e.  RR
17 resubcl 9111 . . 3  |-  ( ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  e.  RR  /\  1  e.  RR )  ->  (
( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  e.  RR )
1815, 16, 17sylancl 643 . 2  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  e.  RR )
19 jm3.1.d . . 3  |-  ( ph  ->  ( K Yrm  ( N  + 
1 ) )  <_  A )
207, 1, 4, 19jm3.1lem1 27110 . 2  |-  ( ph  ->  ( K ^ N
)  <  A )
219, 3remulcld 8863 . . . 4  |-  ( ph  ->  ( A  x.  K
)  e.  RR )
22 resubcl 9111 . . . . 5  |-  ( ( K  e.  RR  /\  1  e.  RR )  ->  ( K  -  1 )  e.  RR )
233, 16, 22sylancl 643 . . . 4  |-  ( ph  ->  ( K  -  1 )  e.  RR )
2421, 23readdcld 8862 . . 3  |-  ( ph  ->  ( ( A  x.  K )  +  ( K  -  1 ) )  e.  RR )
25 eluz2b1 10289 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  ZZ  /\  1  < 
K ) )
2625simprbi 450 . . . . . 6  |-  ( K  e.  ( ZZ>= `  2
)  ->  1  <  K )
271, 26syl 15 . . . . 5  |-  ( ph  ->  1  <  K )
28 eluz2b2 10290 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
2928simplbi 446 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN )
307, 29syl 15 . . . . . . 7  |-  ( ph  ->  A  e.  NN )
3130nngt0d 9789 . . . . . 6  |-  ( ph  ->  0  <  A )
32 ltmulgt11 9616 . . . . . 6  |-  ( ( A  e.  RR  /\  K  e.  RR  /\  0  <  A )  ->  (
1  <  K  <->  A  <  ( A  x.  K ) ) )
339, 3, 31, 32syl3anc 1182 . . . . 5  |-  ( ph  ->  ( 1  <  K  <->  A  <  ( A  x.  K ) ) )
3427, 33mpbid 201 . . . 4  |-  ( ph  ->  A  <  ( A  x.  K ) )
35 uz2m1nn 10292 . . . . . . 7  |-  ( K  e.  ( ZZ>= `  2
)  ->  ( K  -  1 )  e.  NN )
361, 35syl 15 . . . . . 6  |-  ( ph  ->  ( K  -  1 )  e.  NN )
3736nnrpd 10389 . . . . 5  |-  ( ph  ->  ( K  -  1 )  e.  RR+ )
3821, 37ltaddrpd 10419 . . . 4  |-  ( ph  ->  ( A  x.  K
)  <  ( ( A  x.  K )  +  ( K  - 
1 ) ) )
399, 21, 24, 34, 38lttrd 8977 . . 3  |-  ( ph  ->  A  <  ( ( A  x.  K )  +  ( K  - 
1 ) ) )
40 peano2re 8985 . . . . . . 7  |-  ( K  e.  RR  ->  ( K  +  1 )  e.  RR )
413, 40syl 15 . . . . . 6  |-  ( ph  ->  ( K  +  1 )  e.  RR )
4241, 3remulcld 8863 . . . . 5  |-  ( ph  ->  ( ( K  + 
1 )  x.  K
)  e.  RR )
43 resubcl 9111 . . . . . . 7  |-  ( ( ( A  x.  K
)  e.  RR  /\  1  e.  RR )  ->  ( ( A  x.  K )  -  1 )  e.  RR )
4421, 16, 43sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( A  x.  K )  -  1 )  e.  RR )
4544, 14resubcld 9211 . . . . 5  |-  ( ph  ->  ( ( ( A  x.  K )  - 
1 )  -  ( K ^ 2 ) )  e.  RR )
463recnd 8861 . . . . . . . . . 10  |-  ( ph  ->  K  e.  CC )
4746exp1d 11240 . . . . . . . . 9  |-  ( ph  ->  ( K ^ 1 )  =  K )
48 eluz2b2 10290 . . . . . . . . . . . . 13  |-  ( K  e.  ( ZZ>= `  2
)  <->  ( K  e.  NN  /\  1  < 
K ) )
4948simplbi 446 . . . . . . . . . . . 12  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  NN )
501, 49syl 15 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  NN )
5150nnge1d 9788 . . . . . . . . . 10  |-  ( ph  ->  1  <_  K )
52 nnuz 10263 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
534, 52syl6eleq 2373 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
543, 51, 53leexp2ad 11277 . . . . . . . . 9  |-  ( ph  ->  ( K ^ 1 )  <_  ( K ^ N ) )
5547, 54eqbrtrrd 4045 . . . . . . . 8  |-  ( ph  ->  K  <_  ( K ^ N ) )
563, 6, 9, 55, 20lelttrd 8974 . . . . . . 7  |-  ( ph  ->  K  <  A )
57 eluzelz 10238 . . . . . . . . 9  |-  ( K  e.  ( ZZ>= `  2
)  ->  K  e.  ZZ )
581, 57syl 15 . . . . . . . 8  |-  ( ph  ->  K  e.  ZZ )
59 eluzelz 10238 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
607, 59syl 15 . . . . . . . 8  |-  ( ph  ->  A  e.  ZZ )
61 zltp1le 10067 . . . . . . . 8  |-  ( ( K  e.  ZZ  /\  A  e.  ZZ )  ->  ( K  <  A  <->  ( K  +  1 )  <_  A ) )
6258, 60, 61syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( K  <  A  <->  ( K  +  1 )  <_  A ) )
6356, 62mpbid 201 . . . . . 6  |-  ( ph  ->  ( K  +  1 )  <_  A )
6450nngt0d 9789 . . . . . . 7  |-  ( ph  ->  0  <  K )
65 lemul1 9608 . . . . . . 7  |-  ( ( ( K  +  1 )  e.  RR  /\  A  e.  RR  /\  ( K  e.  RR  /\  0  <  K ) )  -> 
( ( K  + 
1 )  <_  A  <->  ( ( K  +  1 )  x.  K )  <_  ( A  x.  K ) ) )
6641, 9, 3, 64, 65syl112anc 1186 . . . . . 6  |-  ( ph  ->  ( ( K  + 
1 )  <_  A  <->  ( ( K  +  1 )  x.  K )  <_  ( A  x.  K ) ) )
6763, 66mpbid 201 . . . . 5  |-  ( ph  ->  ( ( K  + 
1 )  x.  K
)  <_  ( A  x.  K ) )
6842, 21, 45, 67leadd1dd 9386 . . . 4  |-  ( ph  ->  ( ( ( K  +  1 )  x.  K )  +  ( ( ( A  x.  K )  -  1 )  -  ( K ^ 2 ) ) )  <_  ( ( A  x.  K )  +  ( ( ( A  x.  K )  -  1 )  -  ( K ^ 2 ) ) ) )
6921recnd 8861 . . . . . 6  |-  ( ph  ->  ( A  x.  K
)  e.  CC )
7042, 14resubcld 9211 . . . . . . 7  |-  ( ph  ->  ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  e.  RR )
7170recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  e.  CC )
72 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
7372a1i 10 . . . . . 6  |-  ( ph  ->  1  e.  CC )
7469, 71, 73addsub12d 9180 . . . . 5  |-  ( ph  ->  ( ( A  x.  K )  +  ( ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  -  1 ) )  =  ( ( ( ( K  +  1 )  x.  K )  -  ( K ^
2 ) )  +  ( ( A  x.  K )  -  1 ) ) )
7546, 73, 46adddird 8860 . . . . . . . . 9  |-  ( ph  ->  ( ( K  + 
1 )  x.  K
)  =  ( ( K  x.  K )  +  ( 1  x.  K ) ) )
7646sqvald 11242 . . . . . . . . 9  |-  ( ph  ->  ( K ^ 2 )  =  ( K  x.  K ) )
7775, 76oveq12d 5876 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  =  ( ( ( K  x.  K )  +  ( 1  x.  K ) )  -  ( K  x.  K
) ) )
7846, 46mulcld 8855 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  K
)  e.  CC )
79 mulcl 8821 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  K  e.  CC )  ->  ( 1  x.  K
)  e.  CC )
8072, 46, 79sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( 1  x.  K
)  e.  CC )
8178, 80pncan2d 9159 . . . . . . . 8  |-  ( ph  ->  ( ( ( K  x.  K )  +  ( 1  x.  K
) )  -  ( K  x.  K )
)  =  ( 1  x.  K ) )
8246mulid2d 8853 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  K
)  =  K )
8377, 81, 823eqtrd 2319 . . . . . . 7  |-  ( ph  ->  ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  =  K )
8483oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  -  1 )  =  ( K  - 
1 ) )
8584oveq2d 5874 . . . . 5  |-  ( ph  ->  ( ( A  x.  K )  +  ( ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  -  1 ) )  =  ( ( A  x.  K )  +  ( K  -  1 ) ) )
8642recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( K  + 
1 )  x.  K
)  e.  CC )
8714recnd 8861 . . . . . 6  |-  ( ph  ->  ( K ^ 2 )  e.  CC )
8844recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( A  x.  K )  -  1 )  e.  CC )
8986, 87, 88subadd23d 9179 . . . . 5  |-  ( ph  ->  ( ( ( ( K  +  1 )  x.  K )  -  ( K ^ 2 ) )  +  ( ( A  x.  K )  -  1 ) )  =  ( ( ( K  +  1 )  x.  K )  +  ( ( ( A  x.  K )  - 
1 )  -  ( K ^ 2 ) ) ) )
9074, 85, 893eqtr3d 2323 . . . 4  |-  ( ph  ->  ( ( A  x.  K )  +  ( K  -  1 ) )  =  ( ( ( K  +  1 )  x.  K )  +  ( ( ( A  x.  K )  -  1 )  -  ( K ^ 2 ) ) ) )
91 2cn 9816 . . . . . . . . . 10  |-  2  e.  CC
9291a1i 10 . . . . . . . . 9  |-  ( ph  ->  2  e.  CC )
939recnd 8861 . . . . . . . . 9  |-  ( ph  ->  A  e.  CC )
9492, 93, 46mulassd 8858 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  A )  x.  K
)  =  ( 2  x.  ( A  x.  K ) ) )
95692timesd 9954 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( A  x.  K )
)  =  ( ( A  x.  K )  +  ( A  x.  K ) ) )
9694, 95eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( 2  x.  A )  x.  K
)  =  ( ( A  x.  K )  +  ( A  x.  K ) ) )
9796oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  =  ( ( ( A  x.  K )  +  ( A  x.  K ) )  -  ( K ^ 2 ) ) )
9897oveq1d 5873 . . . . 5  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  =  ( ( ( ( A  x.  K
)  +  ( A  x.  K ) )  -  ( K ^
2 ) )  - 
1 ) )
9921, 21readdcld 8862 . . . . . . 7  |-  ( ph  ->  ( ( A  x.  K )  +  ( A  x.  K ) )  e.  RR )
10099recnd 8861 . . . . . 6  |-  ( ph  ->  ( ( A  x.  K )  +  ( A  x.  K ) )  e.  CC )
101100, 87, 73sub32d 9189 . . . . 5  |-  ( ph  ->  ( ( ( ( A  x.  K )  +  ( A  x.  K ) )  -  ( K ^ 2 ) )  -  1 )  =  ( ( ( ( A  x.  K
)  +  ( A  x.  K ) )  -  1 )  -  ( K ^ 2 ) ) )
10269, 69, 73addsubassd 9177 . . . . . . 7  |-  ( ph  ->  ( ( ( A  x.  K )  +  ( A  x.  K
) )  -  1 )  =  ( ( A  x.  K )  +  ( ( A  x.  K )  - 
1 ) ) )
103102oveq1d 5873 . . . . . 6  |-  ( ph  ->  ( ( ( ( A  x.  K )  +  ( A  x.  K ) )  - 
1 )  -  ( K ^ 2 ) )  =  ( ( ( A  x.  K )  +  ( ( A  x.  K )  - 
1 ) )  -  ( K ^ 2 ) ) )
10469, 88, 87addsubassd 9177 . . . . . 6  |-  ( ph  ->  ( ( ( A  x.  K )  +  ( ( A  x.  K )  -  1 ) )  -  ( K ^ 2 ) )  =  ( ( A  x.  K )  +  ( ( ( A  x.  K )  - 
1 )  -  ( K ^ 2 ) ) ) )
105103, 104eqtrd 2315 . . . . 5  |-  ( ph  ->  ( ( ( ( A  x.  K )  +  ( A  x.  K ) )  - 
1 )  -  ( K ^ 2 ) )  =  ( ( A  x.  K )  +  ( ( ( A  x.  K )  - 
1 )  -  ( K ^ 2 ) ) ) )
10698, 101, 1053eqtrd 2319 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  K )  -  ( K ^ 2 ) )  -  1 )  =  ( ( A  x.  K )  +  ( ( ( A  x.  K )  - 
1 )  -  ( K ^ 2 ) ) ) )
10768, 90, 1063brtr4d 4053 . . 3  |-  ( ph  ->  ( ( A  x.  K )  +  ( K  -  1 ) )  <_  ( (
( ( 2  x.  A )  x.  K
)  -  ( K ^ 2 ) )  -  1 ) )
1089, 24, 18, 39, 107ltletrd 8976 . 2  |-  ( ph  ->  A  <  ( ( ( ( 2  x.  A )  x.  K
)  -  ( K ^ 2 ) )  -  1 ) )
1096, 9, 18, 20, 108lttrd 8977 1  |-  ( ph  ->  ( K ^ N
)  <  ( (
( ( 2  x.  A )  x.  K
)  -  ( K ^ 2 ) )  -  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104   Yrm crmy 26986
This theorem is referenced by:  jm3.1lem3  27112  jm3.1  27113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-squarenn 26926  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929  df-pellfund 26930  df-rmx 26987  df-rmy 26988
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