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Theorem aaliou3lem1 19738
Description: Lemma for aaliou3 19747. (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 5881 . . . . . 6  |-  ( c  =  B  ->  (
c  -  A )  =  ( B  -  A ) )
21oveq2d 5890 . . . . 5  |-  ( c  =  B  ->  (
( 1  /  2
) ^ ( c  -  A ) )  =  ( ( 1  /  2 ) ^
( B  -  A
) ) )
32oveq2d 5890 . . . 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 5899 . . . 4  |-  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) )  e.  _V
63, 4, 5fvmpt 5618 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( G `  B )  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
76adantl 452 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  =  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) ) )
8 2rp 10375 . . . . 5  |-  2  e.  RR+
9 simpl 443 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN )
109nnnn0d 10034 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN0 )
11 faccl 11314 . . . . . . . 8  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
1210, 11syl 15 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  NN )
1312nnzd 10132 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  ZZ )
1413znegcld 10135 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  -u ( ! `  A
)  e.  ZZ )
15 rpexpcl 11138 . . . . 5  |-  ( ( 2  e.  RR+  /\  -u ( ! `  A )  e.  ZZ )  ->  (
2 ^ -u ( ! `  A )
)  e.  RR+ )
168, 14, 15sylancr 644 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( 2 ^ -u ( ! `  A )
)  e.  RR+ )
17 2re 9831 . . . . . . 7  |-  2  e.  RR
18 2ne0 9845 . . . . . . 7  |-  2  =/=  0
1917, 18rereccli 9541 . . . . . 6  |-  ( 1  /  2 )  e.  RR
20 halfgt0 9948 . . . . . 6  |-  0  <  ( 1  /  2
)
2119, 20elrpii 10373 . . . . 5  |-  ( 1  /  2 )  e.  RR+
22 eluzelz 10254 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
23 nnz 10061 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  ZZ )
24 zsubcl 10077 . . . . . 6  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ )  ->  ( B  -  A
)  e.  ZZ )
2522, 23, 24syl2anr 464 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( B  -  A
)  e.  ZZ )
26 rpexpcl 11138 . . . . 5  |-  ( ( ( 1  /  2
)  e.  RR+  /\  ( B  -  A )  e.  ZZ )  ->  (
( 1  /  2
) ^ ( B  -  A ) )  e.  RR+ )
2721, 25, 26sylancr 644 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 1  / 
2 ) ^ ( B  -  A )
)  e.  RR+ )
2816, 27rpmulcld 10422 . . 3  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR+ )
2928rpred 10406 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR )
307, 29eqeltrd 2370 1  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( G `  B
)  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   RRcr 8752   1c1 8754    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   ^cexp 11120   !cfa 11304
This theorem is referenced by:  aaliou3lem2  19739  aaliou3lem3  19740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-fac 11305
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