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Mirrors > Home > MPE Home > Th. List > expclzlem | Structured version Visualization version GIF version |
Description: Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
expclzlem | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4260 | . . . 4 ⊢ (𝐴 ∈ (ℂ ∖ {0}) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) | |
2 | difss 3699 | . . . . . 6 ⊢ (ℂ ∖ {0}) ⊆ ℂ | |
3 | eldifsn 4260 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | |
4 | eldifsn 4260 | . . . . . . 7 ⊢ (𝑦 ∈ (ℂ ∖ {0}) ↔ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) | |
5 | mulcl 9899 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
6 | 5 | ad2ant2r 779 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (𝑥 · 𝑦) ∈ ℂ) |
7 | mulne0 10548 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (𝑥 · 𝑦) ≠ 0) | |
8 | eldifsn 4260 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ (ℂ ∖ {0}) ↔ ((𝑥 · 𝑦) ∈ ℂ ∧ (𝑥 · 𝑦) ≠ 0)) | |
9 | 6, 7, 8 | sylanbrc 695 | . . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
10 | 3, 4, 9 | syl2anb 495 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑦 ∈ (ℂ ∖ {0})) → (𝑥 · 𝑦) ∈ (ℂ ∖ {0})) |
11 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
12 | ax-1ne0 9884 | . . . . . . 7 ⊢ 1 ≠ 0 | |
13 | eldifsn 4260 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ 1 ≠ 0)) | |
14 | 11, 12, 13 | mpbir2an 957 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ {0}) |
15 | reccl 10571 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ ℂ) | |
16 | recne0 10577 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (1 / 𝑥) ≠ 0) | |
17 | 15, 16 | jca 553 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) ≠ 0)) |
18 | eldifsn 4260 | . . . . . . . 8 ⊢ ((1 / 𝑥) ∈ (ℂ ∖ {0}) ↔ ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) ≠ 0)) | |
19 | 17, 3, 18 | 3imtr4i 280 | . . . . . . 7 ⊢ (𝑥 ∈ (ℂ ∖ {0}) → (1 / 𝑥) ∈ (ℂ ∖ {0})) |
20 | 19 | adantr 480 | . . . . . 6 ⊢ ((𝑥 ∈ (ℂ ∖ {0}) ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ (ℂ ∖ {0})) |
21 | 2, 10, 14, 20 | expcl2lem 12734 | . . . . 5 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0})) |
22 | 21 | 3expia 1259 | . . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐴 ≠ 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ (ℂ ∖ {0}))) |
23 | 1, 22 | sylanbr 489 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐴 ≠ 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ (ℂ ∖ {0}))) |
24 | 23 | anabss3 860 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ (ℂ ∖ {0}))) |
25 | 24 | 3impia 1253 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 / cdiv 10563 ℤcz 11254 ↑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-rmo 2904 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-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: expclz 12747 expne0i 12754 expghm 19663 |
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