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Theorem aaliou3lem1 22465
Description: Lemma for aaliou3 22474. (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 6282 . . . . . 6  |-  ( c  =  B  ->  (
c  -  A )  =  ( B  -  A ) )
21oveq2d 6291 . . . . 5  |-  ( c  =  B  ->  (
( 1  /  2
) ^ ( c  -  A ) )  =  ( ( 1  /  2 ) ^
( B  -  A
) ) )
32oveq2d 6291 . . . 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 6300 . . . 4  |-  ( ( 2 ^ -u ( ! `  A )
)  x.  ( ( 1  /  2 ) ^ ( B  -  A ) ) )  e.  _V
63, 4, 5fvmpt 5941 . . 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 11214 . . . . 5  |-  2  e.  RR+
9 simpl 457 . . . . . . . . 9  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN )
109nnnn0d 10841 . . . . . . . 8  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  A  e.  NN0 )
11 faccl 12318 . . . . . . . 8  |-  ( A  e.  NN0  ->  ( ! `
 A )  e.  NN )
1210, 11syl 16 . . . . . . 7  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  NN )
1312nnzd 10954 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ! `  A
)  e.  ZZ )
1413znegcld 10957 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  ->  -u ( ! `  A
)  e.  ZZ )
15 rpexpcl 12141 . . . . 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 10743 . . . . . 6  |-  ( 1  /  2 )  e.  RR
18 halfgt0 10745 . . . . . 6  |-  0  <  ( 1  /  2
)
1917, 18elrpii 11212 . . . . 5  |-  ( 1  /  2 )  e.  RR+
20 eluzelz 11080 . . . . . 6  |-  ( B  e.  ( ZZ>= `  A
)  ->  B  e.  ZZ )
21 nnz 10875 . . . . . 6  |-  ( A  e.  NN  ->  A  e.  ZZ )
22 zsubcl 10894 . . . . . 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 12141 . . . . 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 11261 . . 3  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR+ )
2726rpred 11245 . 2  |-  ( ( A  e.  NN  /\  B  e.  ( ZZ>= `  A ) )  -> 
( ( 2 ^
-u ( ! `  A ) )  x.  ( ( 1  / 
2 ) ^ ( B  -  A )
) )  e.  RR )
287, 27eqeltrd 2548 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 1374    e. wcel 1762    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275   RRcr 9480   1c1 9482    x. cmul 9486    - cmin 9794   -ucneg 9795    / cdiv 10195   NNcn 10525   2c2 10574   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209   ^cexp 12122   !cfa 12308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-seq 12064  df-exp 12123  df-fac 12309
This theorem is referenced by:  aaliou3lem2  22466  aaliou3lem3  22467
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