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Theorem aaliou3lem1 20261
Description: Lemma for aaliou3 20270. (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 6090 . . . . . 6  |-  ( c  =  B  ->  (
c  -  A )  =  ( B  -  A ) )
21oveq2d 6099 . . . . 5  |-  ( c  =  B  ->  (
( 1  /  2
) ^ ( c  -  A ) )  =  ( ( 1  /  2 ) ^
( B  -  A
) ) )
32oveq2d 6099 . . . 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 6108 . . . 4  |-  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) )  e.  _V
63, 4, 5fvmpt 5808 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( G `  B )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
76adantl 454 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
8 2rp 10619 . . . . 5  |-  2  e.  RR+
9 simpl 445 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN )
109nnnn0d 10276 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN0 )
11 faccl 11578 . . . . . . . 8  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  NN )
1312nnzd 10376 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  ZZ )
1413znegcld 10379 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  -u ( ! `  A
)  e.  ZZ )
15 rpexpcl 11402 . . . . 5  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
168, 14, 15sylancr 646 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( 2 ^ -u ( ! `  A )
)  e.  RR+ )
17 2re 10071 . . . . . . 7  |-  2  e.  RR
18 2ne0 10085 . . . . . . 7  |-  2  =/=  0
1917, 18rereccli 9781 . . . . . 6  |-  ( 1  /  2 )  e.  RR
20 halfgt0 10190 . . . . . 6  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10617 . . . . 5  |-  ( 1  /  2 )  e.  RR+
22 eluzelz 10498 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
23 nnz 10305 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  ZZ )
24 zsubcl 10321 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  -  A
)  e.  ZZ )
2522, 23, 24syl2anr 466 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( B  -  A
)  e.  ZZ )
26 rpexpcl 11402 . . . . 5  |-  ( ( ( 1  /  2
)  e.  RR+  /\  ( B  -  A )  e.  ZZ )  ->  (
( 1  /  2
) ^ ( B  -  A ) )  e.  RR+ )
2721, 25, 26sylancr 646 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 1  / 
2 ) ^ ( B  -  A )
)  e.  RR+ )
2816, 27rpmulcld 10666 . . 3  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR+ )
2928rpred 10650 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR )
307, 29eqeltrd 2512 1  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   RRcr 8991   1c1 8993    x. cmul 8997    - cmin 9293   -ucneg 9294    / cdiv 9679   NNcn 10002   2c2 10051   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   ^cexp 11384   !cfa 11568
This theorem is referenced by:  aaliou3lem2  20262  aaliou3lem3  20263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-seq 11326  df-exp 11385  df-fac 11569
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