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Theorem aaliou3lem1 21788
Description: Lemma for aaliou3 21797. (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 6093 . . . . . 6  |-  ( c  =  B  ->  (
c  -  A )  =  ( B  -  A ) )
21oveq2d 6102 . . . . 5  |-  ( c  =  B  ->  (
( 1  /  2
) ^ ( c  -  A ) )  =  ( ( 1  /  2 ) ^
( B  -  A
) ) )
32oveq2d 6102 . . . 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 6111 . . . 4  |-  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) )  e.  _V
63, 4, 5fvmpt 5769 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( G `  B )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
76adantl 466 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
8 2rp 10988 . . . . 5  |-  2  e.  RR+
9 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN )
109nnnn0d 10628 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN0 )
11 faccl 12053 . . . . . . . 8  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  NN )
1312nnzd 10738 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  ZZ )
1413znegcld 10741 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  -u ( ! `  A
)  e.  ZZ )
15 rpexpcl 11876 . . . . 5  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
168, 14, 15sylancr 663 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( 2 ^ -u ( ! `  A )
)  e.  RR+ )
17 halfre 10532 . . . . . 6  |-  ( 1  /  2 )  e.  RR
18 halfgt0 10534 . . . . . 6  |-  0  <  ( 1  /  2
)
1917, 18elrpii 10986 . . . . 5  |-  ( 1  /  2 )  e.  RR+
20 eluzelz 10862 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
21 nnz 10660 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  ZZ )
22 zsubcl 10679 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  -  A
)  e.  ZZ )
2320, 21, 22syl2anr 478 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( B  -  A
)  e.  ZZ )
24 rpexpcl 11876 . . . . 5  |-  ( ( ( 1  /  2
)  e.  RR+  /\  ( B  -  A )  e.  ZZ )  ->  (
( 1  /  2
) ^ ( B  -  A ) )  e.  RR+ )
2519, 23, 24sylancr 663 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 1  / 
2 ) ^ ( B  -  A )
)  e.  RR+ )
2616, 25rpmulcld 11035 . . 3  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR+ )
2726rpred 11019 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR )
287, 27eqeltrd 2512 1  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    e. cmpt 4345   ` cfv 5413  (class class class)co 6086   RRcr 9273   1c1 9275    x. cmul 9279    - cmin 9587   -ucneg 9588    / cdiv 9985   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   ^cexp 11857   !cfa 12043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-seq 11799  df-exp 11858  df-fac 12044
This theorem is referenced by:  aaliou3lem2  21789  aaliou3lem3  21790
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