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Mirrors > Home > MPE Home > Th. List > aaliou3lem1 | Structured version Visualization version GIF version |
Description: Lemma for aaliou3 24942. (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 7165 | . . . . . 6 ⊢ (𝑐 = 𝐵 → (𝑐 − 𝐴) = (𝐵 − 𝐴)) | |
2 | 1 | oveq2d 7174 | . . . . 5 ⊢ (𝑐 = 𝐵 → ((1 / 2)↑(𝑐 − 𝐴)) = ((1 / 2)↑(𝐵 − 𝐴))) |
3 | 2 | oveq2d 7174 | . . . 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 7191 | . . . 4 ⊢ ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ V | |
6 | 3, 4, 5 | fvmpt 6770 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
7 | 6 | adantl 484 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) = ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴)))) |
8 | 2rp 12397 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
9 | simpl 485 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ) | |
10 | 9 | nnnn0d 11958 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐴 ∈ ℕ0) |
11 | 10 | faccld 13647 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℕ) |
12 | 11 | nnzd 12089 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (!‘𝐴) ∈ ℤ) |
13 | 12 | znegcld 12092 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → -(!‘𝐴) ∈ ℤ) |
14 | rpexpcl 13451 | . . . . 5 ⊢ ((2 ∈ ℝ+ ∧ -(!‘𝐴) ∈ ℤ) → (2↑-(!‘𝐴)) ∈ ℝ+) | |
15 | 8, 13, 14 | sylancr 589 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (2↑-(!‘𝐴)) ∈ ℝ+) |
16 | halfre 11854 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
17 | halfgt0 11856 | . . . . . 6 ⊢ 0 < (1 / 2) | |
18 | 16, 17 | elrpii 12395 | . . . . 5 ⊢ (1 / 2) ∈ ℝ+ |
19 | eluzelz 12256 | . . . . . 6 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
20 | nnz 12007 | . . . . . 6 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ) | |
21 | zsubcl 12027 | . . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐵 − 𝐴) ∈ ℤ) | |
22 | 19, 20, 21 | syl2anr 598 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐵 − 𝐴) ∈ ℤ) |
23 | rpexpcl 13451 | . . . . 5 ⊢ (((1 / 2) ∈ ℝ+ ∧ (𝐵 − 𝐴) ∈ ℤ) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) | |
24 | 18, 22, 23 | sylancr 589 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((1 / 2)↑(𝐵 − 𝐴)) ∈ ℝ+) |
25 | 15, 24 | rpmulcld 12450 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ+) |
26 | 25 | rpred 12434 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → ((2↑-(!‘𝐴)) · ((1 / 2)↑(𝐵 − 𝐴))) ∈ ℝ) |
27 | 7, 26 | eqeltrd 2915 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → (𝐺‘𝐵) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 1c1 10540 · cmul 10544 − cmin 10872 -cneg 10873 / cdiv 11299 ℕcn 11640 2c2 11695 ℤcz 11984 ℤ≥cuz 12246 ℝ+crp 12392 ↑cexp 13432 !cfa 13636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-fac 13637 |
This theorem is referenced by: aaliou3lem2 24934 aaliou3lem3 24935 |
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