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Mirrors > Home > MPE Home > Th. List > aaliou3lem1 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 25454. (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 7267 | . . . . . 6 ⊢ (𝑐 = 𝐵 → (𝑐 − 𝐴) = (𝐵 − 𝐴)) | |
2 | 1 | oveq2d 7276 | . . . . 5 ⊢ (𝑐 = 𝐵 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐵 − 𝐴))) |
3 | 2 | oveq2d 7276 | . . . 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 7293 | . . . 4 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ V | |
6 | 3, 4, 5 | fvmpt 6862 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
7 | 6 | adantl 481 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
8 | 2rp 12680 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
9 | simpl 482 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ) | |
10 | 9 | nnnn0d 12239 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ0) |
11 | 10 | faccld 13942 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℕ) |
12 | 11 | nnzd 12370 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℤ) |
13 | 12 | znegcld 12373 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → -(!‘𝐴) ∈ ℤ) |
14 | rpexpcl 13745 | . . . . 5 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ+) | |
15 | 8, 13, 14 | sylancr 586 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ+) |
16 | halfre 12133 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
17 | halfgt0 12135 | . . . . . 6 ⊢ 0 < (1 / 2) | |
18 | 16, 17 | elrpii 12678 | . . . . 5 ⊢ (1 / 2) ∈ ℝ+ |
19 | eluzelz 12537 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
20 | nnz 12288 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
21 | zsubcl 12308 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 − 𝐴) ∈ ℤ) | |
22 | 19, 20, 21 | syl2anr 596 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐵 − 𝐴) ∈ ℤ) |
23 | rpexpcl 13745 | . . . . 5 ⊢ (((1 / 2) ∈ ℝ+ ∧ (𝐵 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) | |
24 | 18, 22, 23 | sylancr 586 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) |
25 | 15, 24 | rpmulcld 12733 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ+) |
26 | 25 | rpred 12717 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ) |
27 | 7, 26 | eqeltrd 2837 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ↦ cmpt 5158 ‘cfv 6423 (class class class)co 7260 ℝcr 10817 1c1 10819 · cmul 10823 − cmin 11151 -cneg 11152 / cdiv 11578 ℕcn 11919 2c2 11974 ℤcz 12265 ℤ≥cuz 12527 ℝ+crp 12675 ↑cexp 13726 !cfa 13931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-div 11579 df-nn 11920 df-2 11982 df-n0 12180 df-z 12266 df-uz 12528 df-rp 12676 df-seq 13666 df-exp 13727 df-fac 13932 |
This theorem is referenced by: aaliou3lem2 25446 aaliou3lem3 25447 |
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