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Mirrors > Home > ILE Home > Th. List > expcllem | GIF version |
Description: Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.) |
Ref | Expression |
---|---|
expcllem.1 | ⊢ 𝐹 ⊆ ℂ |
expcllem.2 | ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹) |
expcllem.3 | ⊢ 1 ∈ 𝐹 |
Ref | Expression |
---|---|
expcllem | ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ0) → (A↑B) ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 7959 | . 2 ⊢ (B ∈ ℕ0 ↔ (B ∈ ℕ ∨ B = 0)) | |
2 | oveq2 5463 | . . . . . . 7 ⊢ (z = 1 → (A↑z) = (A↑1)) | |
3 | 2 | eleq1d 2103 | . . . . . 6 ⊢ (z = 1 → ((A↑z) ∈ 𝐹 ↔ (A↑1) ∈ 𝐹)) |
4 | 3 | imbi2d 219 | . . . . 5 ⊢ (z = 1 → ((A ∈ 𝐹 → (A↑z) ∈ 𝐹) ↔ (A ∈ 𝐹 → (A↑1) ∈ 𝐹))) |
5 | oveq2 5463 | . . . . . . 7 ⊢ (z = w → (A↑z) = (A↑w)) | |
6 | 5 | eleq1d 2103 | . . . . . 6 ⊢ (z = w → ((A↑z) ∈ 𝐹 ↔ (A↑w) ∈ 𝐹)) |
7 | 6 | imbi2d 219 | . . . . 5 ⊢ (z = w → ((A ∈ 𝐹 → (A↑z) ∈ 𝐹) ↔ (A ∈ 𝐹 → (A↑w) ∈ 𝐹))) |
8 | oveq2 5463 | . . . . . . 7 ⊢ (z = (w + 1) → (A↑z) = (A↑(w + 1))) | |
9 | 8 | eleq1d 2103 | . . . . . 6 ⊢ (z = (w + 1) → ((A↑z) ∈ 𝐹 ↔ (A↑(w + 1)) ∈ 𝐹)) |
10 | 9 | imbi2d 219 | . . . . 5 ⊢ (z = (w + 1) → ((A ∈ 𝐹 → (A↑z) ∈ 𝐹) ↔ (A ∈ 𝐹 → (A↑(w + 1)) ∈ 𝐹))) |
11 | oveq2 5463 | . . . . . . 7 ⊢ (z = B → (A↑z) = (A↑B)) | |
12 | 11 | eleq1d 2103 | . . . . . 6 ⊢ (z = B → ((A↑z) ∈ 𝐹 ↔ (A↑B) ∈ 𝐹)) |
13 | 12 | imbi2d 219 | . . . . 5 ⊢ (z = B → ((A ∈ 𝐹 → (A↑z) ∈ 𝐹) ↔ (A ∈ 𝐹 → (A↑B) ∈ 𝐹))) |
14 | expcllem.1 | . . . . . . . . 9 ⊢ 𝐹 ⊆ ℂ | |
15 | 14 | sseli 2935 | . . . . . . . 8 ⊢ (A ∈ 𝐹 → A ∈ ℂ) |
16 | exp1 8915 | . . . . . . . 8 ⊢ (A ∈ ℂ → (A↑1) = A) | |
17 | 15, 16 | syl 14 | . . . . . . 7 ⊢ (A ∈ 𝐹 → (A↑1) = A) |
18 | 17 | eleq1d 2103 | . . . . . 6 ⊢ (A ∈ 𝐹 → ((A↑1) ∈ 𝐹 ↔ A ∈ 𝐹)) |
19 | 18 | ibir 166 | . . . . 5 ⊢ (A ∈ 𝐹 → (A↑1) ∈ 𝐹) |
20 | expcllem.2 | . . . . . . . . . . . 12 ⊢ ((x ∈ 𝐹 ∧ y ∈ 𝐹) → (x · y) ∈ 𝐹) | |
21 | 20 | caovcl 5597 | . . . . . . . . . . 11 ⊢ (((A↑w) ∈ 𝐹 ∧ A ∈ 𝐹) → ((A↑w) · A) ∈ 𝐹) |
22 | 21 | ancoms 255 | . . . . . . . . . 10 ⊢ ((A ∈ 𝐹 ∧ (A↑w) ∈ 𝐹) → ((A↑w) · A) ∈ 𝐹) |
23 | 22 | adantlr 446 | . . . . . . . . 9 ⊢ (((A ∈ 𝐹 ∧ w ∈ ℕ) ∧ (A↑w) ∈ 𝐹) → ((A↑w) · A) ∈ 𝐹) |
24 | nnnn0 7964 | . . . . . . . . . . . 12 ⊢ (w ∈ ℕ → w ∈ ℕ0) | |
25 | expp1 8916 | . . . . . . . . . . . 12 ⊢ ((A ∈ ℂ ∧ w ∈ ℕ0) → (A↑(w + 1)) = ((A↑w) · A)) | |
26 | 15, 24, 25 | syl2an 273 | . . . . . . . . . . 11 ⊢ ((A ∈ 𝐹 ∧ w ∈ ℕ) → (A↑(w + 1)) = ((A↑w) · A)) |
27 | 26 | eleq1d 2103 | . . . . . . . . . 10 ⊢ ((A ∈ 𝐹 ∧ w ∈ ℕ) → ((A↑(w + 1)) ∈ 𝐹 ↔ ((A↑w) · A) ∈ 𝐹)) |
28 | 27 | adantr 261 | . . . . . . . . 9 ⊢ (((A ∈ 𝐹 ∧ w ∈ ℕ) ∧ (A↑w) ∈ 𝐹) → ((A↑(w + 1)) ∈ 𝐹 ↔ ((A↑w) · A) ∈ 𝐹)) |
29 | 23, 28 | mpbird 156 | . . . . . . . 8 ⊢ (((A ∈ 𝐹 ∧ w ∈ ℕ) ∧ (A↑w) ∈ 𝐹) → (A↑(w + 1)) ∈ 𝐹) |
30 | 29 | exp31 346 | . . . . . . 7 ⊢ (A ∈ 𝐹 → (w ∈ ℕ → ((A↑w) ∈ 𝐹 → (A↑(w + 1)) ∈ 𝐹))) |
31 | 30 | com12 27 | . . . . . 6 ⊢ (w ∈ ℕ → (A ∈ 𝐹 → ((A↑w) ∈ 𝐹 → (A↑(w + 1)) ∈ 𝐹))) |
32 | 31 | a2d 23 | . . . . 5 ⊢ (w ∈ ℕ → ((A ∈ 𝐹 → (A↑w) ∈ 𝐹) → (A ∈ 𝐹 → (A↑(w + 1)) ∈ 𝐹))) |
33 | 4, 7, 10, 13, 19, 32 | nnind 7711 | . . . 4 ⊢ (B ∈ ℕ → (A ∈ 𝐹 → (A↑B) ∈ 𝐹)) |
34 | 33 | impcom 116 | . . 3 ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ) → (A↑B) ∈ 𝐹) |
35 | oveq2 5463 | . . . . 5 ⊢ (B = 0 → (A↑B) = (A↑0)) | |
36 | exp0 8913 | . . . . . 6 ⊢ (A ∈ ℂ → (A↑0) = 1) | |
37 | 15, 36 | syl 14 | . . . . 5 ⊢ (A ∈ 𝐹 → (A↑0) = 1) |
38 | 35, 37 | sylan9eqr 2091 | . . . 4 ⊢ ((A ∈ 𝐹 ∧ B = 0) → (A↑B) = 1) |
39 | expcllem.3 | . . . 4 ⊢ 1 ∈ 𝐹 | |
40 | 38, 39 | syl6eqel 2125 | . . 3 ⊢ ((A ∈ 𝐹 ∧ B = 0) → (A↑B) ∈ 𝐹) |
41 | 34, 40 | jaodan 709 | . 2 ⊢ ((A ∈ 𝐹 ∧ (B ∈ ℕ ∨ B = 0)) → (A↑B) ∈ 𝐹) |
42 | 1, 41 | sylan2b 271 | 1 ⊢ ((A ∈ 𝐹 ∧ B ∈ ℕ0) → (A↑B) ∈ 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 (class class class)co 5455 ℂcc 6709 0cc0 6711 1c1 6712 + caddc 6714 · cmul 6716 ℕcn 7695 ℕ0cn0 7957 ↑cexp 8908 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 ax-pre-mulext 6801 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rmo 2308 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-if 3326 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-frec 5918 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-reap 7359 df-ap 7366 df-div 7434 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 df-iseq 8893 df-iexp 8909 |
This theorem is referenced by: expcl2lemap 8921 nnexpcl 8922 nn0expcl 8923 zexpcl 8924 qexpcl 8925 reexpcl 8926 expcl 8927 expge0 8945 expge1 8946 |
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