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Mirrors > Home > ILE Home > Th. List > expclzaplem | GIF version |
Description: Closure law for integer exponentiation. Lemma for expclzap 9501 and expap0i 9508. (Contributed by Jim Kingdon, 9-Jun-2020.) |
Ref | Expression |
---|---|
expclzaplem | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3788 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 # 0 ↔ 𝐴 # 0)) | |
2 | 1 | elrab 2749 | . . . 4 ⊢ (𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝐴 ∈ ℂ ∧ 𝐴 # 0)) |
3 | ssrab2 3079 | . . . . . 6 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ | |
4 | breq1 3788 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 # 0 ↔ 𝑥 # 0)) | |
5 | 4 | elrab 2749 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0)) |
6 | breq1 3788 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 # 0 ↔ 𝑦 # 0)) | |
7 | 6 | elrab 2749 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) |
8 | mulcl 7100 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
9 | 8 | ad2ant2r 492 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) ∈ ℂ) |
10 | mulap0 7744 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) # 0) | |
11 | breq1 3788 | . . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (𝑧 # 0 ↔ (𝑥 · 𝑦) # 0)) | |
12 | 11 | elrab 2749 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ ((𝑥 · 𝑦) ∈ ℂ ∧ (𝑥 · 𝑦) # 0)) |
13 | 9, 10, 12 | sylanbrc 408 | . . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
14 | 5, 7, 13 | syl2anb 285 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝑦 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
15 | ax-1cn 7069 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
16 | 1ap0 7690 | . . . . . . 7 ⊢ 1 # 0 | |
17 | breq1 3788 | . . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧 # 0 ↔ 1 # 0)) | |
18 | 17 | elrab 2749 | . . . . . . 7 ⊢ (1 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (1 ∈ ℂ ∧ 1 # 0)) |
19 | 15, 16, 18 | mpbir2an 883 | . . . . . 6 ⊢ 1 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} |
20 | recclap 7767 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℂ) | |
21 | recap0 7773 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) # 0) | |
22 | 20, 21 | jca 300 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) # 0)) |
23 | breq1 3788 | . . . . . . . . 9 ⊢ (𝑧 = (1 / 𝑥) → (𝑧 # 0 ↔ (1 / 𝑥) # 0)) | |
24 | 23 | elrab 2749 | . . . . . . . 8 ⊢ ((1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) # 0)) |
25 | 22, 5, 24 | 3imtr4i 199 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} → (1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
26 | 25 | adantr 270 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝑥 # 0) → (1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
27 | 3, 14, 19, 26 | expcl2lemap 9488 | . . . . 5 ⊢ ((𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
28 | 27 | 3expia 1140 | . . . 4 ⊢ ((𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
29 | 2, 28 | sylanbr 279 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
30 | 29 | anabss3 549 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
31 | 30 | 3impia 1135 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 ∈ wcel 1433 {crab 2352 class class class wbr 3785 (class class class)co 5532 ℂcc 6979 0cc0 6981 1c1 6982 · cmul 6986 # cap 7681 / cdiv 7760 ℤcz 8351 ↑cexp 9475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 df-iexp 9476 |
This theorem is referenced by: expclzap 9501 expap0i 9508 |
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