Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > expclzaplem | GIF version |
Description: Closure law for integer exponentiation. Lemma for expclzap 10349 and expap0i 10356. (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 3940 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 # 0 ↔ 𝐴 # 0)) | |
2 | 1 | elrab 2844 | . . . 4 ⊢ (𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝐴 ∈ ℂ ∧ 𝐴 # 0)) |
3 | ssrab2 3187 | . . . . . 6 ⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ⊆ ℂ | |
4 | breq1 3940 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 # 0 ↔ 𝑥 # 0)) | |
5 | 4 | elrab 2844 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝑥 ∈ ℂ ∧ 𝑥 # 0)) |
6 | breq1 3940 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝑧 # 0 ↔ 𝑦 # 0)) | |
7 | 6 | elrab 2844 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) |
8 | mulcl 7771 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
9 | 8 | ad2ant2r 501 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) ∈ ℂ) |
10 | mulap0 8439 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) # 0) | |
11 | breq1 3940 | . . . . . . . . 9 ⊢ (𝑧 = (𝑥 · 𝑦) → (𝑧 # 0 ↔ (𝑥 · 𝑦) # 0)) | |
12 | 11 | elrab 2844 | . . . . . . . 8 ⊢ ((𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ ((𝑥 · 𝑦) ∈ ℂ ∧ (𝑥 · 𝑦) # 0)) |
13 | 9, 10, 12 | sylanbrc 414 | . . . . . . 7 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 # 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 # 0)) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
14 | 5, 7, 13 | syl2anb 289 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝑦 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) → (𝑥 · 𝑦) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
15 | ax-1cn 7737 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
16 | 1ap0 8376 | . . . . . . 7 ⊢ 1 # 0 | |
17 | breq1 3940 | . . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧 # 0 ↔ 1 # 0)) | |
18 | 17 | elrab 2844 | . . . . . . 7 ⊢ (1 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ (1 ∈ ℂ ∧ 1 # 0)) |
19 | 15, 16, 18 | mpbir2an 927 | . . . . . 6 ⊢ 1 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} |
20 | recclap 8463 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) ∈ ℂ) | |
21 | recap0 8469 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → (1 / 𝑥) # 0) | |
22 | 20, 21 | jca 304 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 # 0) → ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) # 0)) |
23 | breq1 3940 | . . . . . . . . 9 ⊢ (𝑧 = (1 / 𝑥) → (𝑧 # 0 ↔ (1 / 𝑥) # 0)) | |
24 | 23 | elrab 2844 | . . . . . . . 8 ⊢ ((1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ↔ ((1 / 𝑥) ∈ ℂ ∧ (1 / 𝑥) # 0)) |
25 | 22, 5, 24 | 3imtr4i 200 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} → (1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
26 | 25 | adantr 274 | . . . . . 6 ⊢ ((𝑥 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝑥 # 0) → (1 / 𝑥) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
27 | 3, 14, 19, 26 | expcl2lemap 10336 | . . . . 5 ⊢ ((𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
28 | 27 | 3expia 1184 | . . . 4 ⊢ ((𝐴 ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0} ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
29 | 2, 28 | sylanbr 283 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
30 | 29 | anabss3 575 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑁 ∈ ℤ → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})) |
31 | 30 | 3impia 1179 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 1481 {crab 2421 class class class wbr 3937 (class class class)co 5782 ℂcc 7642 0cc0 7644 1c1 7645 · cmul 7649 # cap 8367 / cdiv 8456 ℤcz 9078 ↑cexp 10323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 df-exp 10324 |
This theorem is referenced by: expclzap 10349 expap0i 10356 |
Copyright terms: Public domain | W3C validator |