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Theorem aaliou3lem1 20212
Description: Lemma for aaliou3 20221. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Hypothesis
Ref Expression
aaliou3lem.a  |-  G  =  ( c  e.  (
ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( c  -  A ) ) ) )
Assertion
Ref Expression
aaliou3lem1  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  e.  RR )
Distinct variable groups:    A, c    B, c
Allowed substitution hint:    G( c)

Proof of Theorem aaliou3lem1
StepHypRef Expression
1 oveq1 6047 . . . . . 6  |-  ( c  =  B  ->  (
c  -  A )  =  ( B  -  A ) )
21oveq2d 6056 . . . . 5  |-  ( c  =  B  ->  (
( 1  /  2
) ^ ( c  -  A ) )  =  ( ( 1  /  2 ) ^
( B  -  A
) ) )
32oveq2d 6056 . . . 4  |-  ( c  =  B  ->  (
( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( c  -  A ) ) )  =  ( ( 2 ^ -u ( ! `
 A ) )  x.  ( ( 1  /  2 ) ^
( B  -  A
) ) ) )
4 aaliou3lem.a . . . 4  |-  G  =  ( c  e.  (
ZZ>= `  A )  |->  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( c  -  A ) ) ) )
5 ovex 6065 . . . 4  |-  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) )  e.  _V
63, 4, 5fvmpt 5765 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( G `  B )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
76adantl 453 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
8 2rp 10573 . . . . 5  |-  2  e.  RR+
9 simpl 444 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN )
109nnnn0d 10230 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN0 )
11 faccl 11531 . . . . . . . 8  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  NN )
1312nnzd 10330 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  ZZ )
1413znegcld 10333 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  -u ( ! `  A
)  e.  ZZ )
15 rpexpcl 11355 . . . . 5  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
168, 14, 15sylancr 645 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( 2 ^ -u ( ! `  A )
)  e.  RR+ )
17 2re 10025 . . . . . . 7  |-  2  e.  RR
18 2ne0 10039 . . . . . . 7  |-  2  =/=  0
1917, 18rereccli 9735 . . . . . 6  |-  ( 1  /  2 )  e.  RR
20 halfgt0 10144 . . . . . 6  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10571 . . . . 5  |-  ( 1  /  2 )  e.  RR+
22 eluzelz 10452 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
23 nnz 10259 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  ZZ )
24 zsubcl 10275 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  -  A
)  e.  ZZ )
2522, 23, 24syl2anr 465 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( B  -  A
)  e.  ZZ )
26 rpexpcl 11355 . . . . 5  |-  ( ( ( 1  /  2
)  e.  RR+  /\  ( B  -  A )  e.  ZZ )  ->  (
( 1  /  2
) ^ ( B  -  A ) )  e.  RR+ )
2721, 25, 26sylancr 645 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 1  / 
2 ) ^ ( B  -  A )
)  e.  RR+ )
2816, 27rpmulcld 10620 . . 3  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR+ )
2928rpred 10604 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR )
307, 29eqeltrd 2478 1  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947    x. cmul 8951    - cmin 9247   -ucneg 9248    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   ^cexp 11337   !cfa 11521
This theorem is referenced by:  aaliou3lem2  20213  aaliou3lem3  20214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-fac 11522
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