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Mirrors > Home > MPE Home > Th. List > expge0 | Structured version Visualization version GIF version |
Description: Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
expge0 | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4587 | . . . . 5 ⊢ (𝑧 = 𝐴 → (0 ≤ 𝑧 ↔ 0 ≤ 𝐴)) | |
2 | 1 | elrab 3331 | . . . 4 ⊢ (𝐴 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) |
3 | ssrab2 3650 | . . . . . . 7 ⊢ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ⊆ ℝ | |
4 | ax-resscn 9872 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
5 | 3, 4 | sstri 3577 | . . . . . 6 ⊢ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ⊆ ℂ |
6 | breq2 4587 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (0 ≤ 𝑧 ↔ 0 ≤ 𝑥)) | |
7 | 6 | elrab 3331 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) |
8 | breq2 4587 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (0 ≤ 𝑧 ↔ 0 ≤ 𝑦)) | |
9 | 8 | elrab 3331 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) |
10 | remulcl 9900 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ) | |
11 | 10 | ad2ant2r 779 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑥 · 𝑦) ∈ ℝ) |
12 | mulge0 10425 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → 0 ≤ (𝑥 · 𝑦)) | |
13 | breq2 4587 | . . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (0 ≤ 𝑧 ↔ 0 ≤ (𝑥 · 𝑦))) | |
14 | 13 | elrab 3331 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ ((𝑥 · 𝑦) ∈ ℝ ∧ 0 ≤ (𝑥 · 𝑦))) |
15 | 11, 12, 14 | sylanbrc 695 | . . . . . . 7 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (𝑦 ∈ ℝ ∧ 0 ≤ 𝑦)) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧}) |
16 | 7, 9, 15 | syl2anb 495 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ∧ 𝑦 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧}) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧}) |
17 | 1re 9918 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
18 | 0le1 10430 | . . . . . . 7 ⊢ 0 ≤ 1 | |
19 | breq2 4587 | . . . . . . . 8 ⊢ (𝑧 = 1 → (0 ≤ 𝑧 ↔ 0 ≤ 1)) | |
20 | 19 | elrab 3331 | . . . . . . 7 ⊢ (1 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ (1 ∈ ℝ ∧ 0 ≤ 1)) |
21 | 17, 18, 20 | mpbir2an 957 | . . . . . 6 ⊢ 1 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} |
22 | 5, 16, 21 | expcllem 12733 | . . . . 5 ⊢ ((𝐴 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧}) |
23 | breq2 4587 | . . . . . . 7 ⊢ (𝑧 = (𝐴↑𝑁) → (0 ≤ 𝑧 ↔ 0 ≤ (𝐴↑𝑁))) | |
24 | 23 | elrab 3331 | . . . . . 6 ⊢ ((𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ↔ ((𝐴↑𝑁) ∈ ℝ ∧ 0 ≤ (𝐴↑𝑁))) |
25 | 24 | simprbi 479 | . . . . 5 ⊢ ((𝐴↑𝑁) ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} → 0 ≤ (𝐴↑𝑁)) |
26 | 22, 25 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ {𝑧 ∈ ℝ ∣ 0 ≤ 𝑧} ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝐴↑𝑁)) |
27 | 2, 26 | sylanbr 489 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝐴↑𝑁)) |
28 | 27 | 3impa 1251 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝐴↑𝑁)) |
29 | 28 | 3com23 1263 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 {crab 2900 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 ≤ cle 9954 ℕ0cn0 11169 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: leexp2r 12780 leexp1a 12781 expge0d 12888 rpnnen2lem4 14785 |
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