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Mirrors > Home > MPE Home > Th. List > aaliou3lem1 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 24106. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
Ref | Expression |
---|---|
aaliou3lem.a | ⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴)))) |
Ref | Expression |
---|---|
aaliou3lem1 | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6657 | . . . . . 6 ⊢ (𝑐 = 𝐵 → (𝑐 − 𝐴) = (𝐵 − 𝐴)) | |
2 | 1 | oveq2d 6666 | . . . . 5 ⊢ (𝑐 = 𝐵 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐵 − 𝐴))) |
3 | 2 | oveq2d 6666 | . . . 4 ⊢ (𝑐 = 𝐵 → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴))) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
4 | aaliou3lem.a | . . . 4 ⊢ 𝐺 = (𝑐 ∈ (ℤ≥‘𝐴) ↦ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝑐 − 𝐴)))) | |
5 | ovex 6678 | . . . 4 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ V | |
6 | 3, 4, 5 | fvmpt 6282 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
7 | 6 | adantl 482 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
8 | 2rp 11837 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
9 | simpl 473 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ) | |
10 | 9 | nnnn0d 11351 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ0) |
11 | faccl 13070 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) ∈ ℕ) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℕ) |
13 | 12 | nnzd 11481 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℤ) |
14 | 13 | znegcld 11484 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → -(!‘𝐴) ∈ ℤ) |
15 | rpexpcl 12879 | . . . . 5 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ+) | |
16 | 8, 14, 15 | sylancr 695 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ+) |
17 | halfre 11246 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
18 | halfgt0 11248 | . . . . . 6 ⊢ 0 < (1 / 2) | |
19 | 17, 18 | elrpii 11835 | . . . . 5 ⊢ (1 / 2) ∈ ℝ+ |
20 | eluzelz 11697 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
21 | nnz 11399 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
22 | zsubcl 11419 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 − 𝐴) ∈ ℤ) | |
23 | 20, 21, 22 | syl2anr 495 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐵 − 𝐴) ∈ ℤ) |
24 | rpexpcl 12879 | . . . . 5 ⊢ (((1 / 2) ∈ ℝ+ ∧ (𝐵 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) | |
25 | 19, 23, 24 | sylancr 695 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) |
26 | 16, 25 | rpmulcld 11888 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ+) |
27 | 26 | rpred 11872 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ) |
28 | 7, 27 | eqeltrd 2701 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 · cmul 9941 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ℤcz 11377 ℤ≥cuz 11687 ℝ+crp 11832 ↑cexp 12860 !cfa 13060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-fac 13061 |
This theorem is referenced by: aaliou3lem2 24098 aaliou3lem3 24099 |
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