Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnsrexpcl | Structured version Visualization version GIF version |
Description: Exponentiation is closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
cnsrexpcl.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) |
cnsrexpcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
cnsrexpcl.y | ⊢ (𝜑 → 𝑌 ∈ ℕ0) |
Ref | Expression |
---|---|
cnsrexpcl | ⊢ (𝜑 → (𝑋↑𝑌) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsrexpcl.y | . 2 ⊢ (𝜑 → 𝑌 ∈ ℕ0) | |
2 | oveq2 7163 | . . . . 5 ⊢ (𝑎 = 0 → (𝑋↑𝑎) = (𝑋↑0)) | |
3 | 2 | eleq1d 2897 | . . . 4 ⊢ (𝑎 = 0 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑0) ∈ 𝑆)) |
4 | 3 | imbi2d 343 | . . 3 ⊢ (𝑎 = 0 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑0) ∈ 𝑆))) |
5 | oveq2 7163 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑋↑𝑎) = (𝑋↑𝑏)) | |
6 | 5 | eleq1d 2897 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑏) ∈ 𝑆)) |
7 | 6 | imbi2d 343 | . . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑏) ∈ 𝑆))) |
8 | oveq2 7163 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝑋↑𝑎) = (𝑋↑(𝑏 + 1))) | |
9 | 8 | eleq1d 2897 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑(𝑏 + 1)) ∈ 𝑆)) |
10 | 9 | imbi2d 343 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
11 | oveq2 7163 | . . . . 5 ⊢ (𝑎 = 𝑌 → (𝑋↑𝑎) = (𝑋↑𝑌)) | |
12 | 11 | eleq1d 2897 | . . . 4 ⊢ (𝑎 = 𝑌 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑌) ∈ 𝑆)) |
13 | 12 | imbi2d 343 | . . 3 ⊢ (𝑎 = 𝑌 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑌) ∈ 𝑆))) |
14 | cnsrexpcl.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
15 | cnfldbas 20548 | . . . . . . . 8 ⊢ ℂ = (Base‘ℂfld) | |
16 | 15 | subrgss 19535 | . . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
17 | 14, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
18 | cnsrexpcl.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
19 | 17, 18 | sseldd 3967 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
20 | 19 | exp0d 13503 | . . . 4 ⊢ (𝜑 → (𝑋↑0) = 1) |
21 | cnfld1 20569 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 21 | subrg1cl 19542 | . . . . 5 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑆) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) |
24 | 20, 23 | eqeltrd 2913 | . . 3 ⊢ (𝜑 → (𝑋↑0) ∈ 𝑆) |
25 | 19 | 3ad2ant2 1130 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ ℂ) |
26 | simp1 1132 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑏 ∈ ℕ0) | |
27 | 25, 26 | expp1d 13510 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) = ((𝑋↑𝑏) · 𝑋)) |
28 | 14 | 3ad2ant2 1130 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑆 ∈ (SubRing‘ℂfld)) |
29 | simp3 1134 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑𝑏) ∈ 𝑆) | |
30 | 18 | 3ad2ant2 1130 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
31 | cnfldmul 20550 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
32 | 31 | subrgmcl 19546 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑋↑𝑏) ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
33 | 28, 29, 30, 32 | syl3anc 1367 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
34 | 27, 33 | eqeltrd 2913 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) ∈ 𝑆) |
35 | 34 | 3exp 1115 | . . . 4 ⊢ (𝑏 ∈ ℕ0 → (𝜑 → ((𝑋↑𝑏) ∈ 𝑆 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
36 | 35 | a2d 29 | . . 3 ⊢ (𝑏 ∈ ℕ0 → ((𝜑 → (𝑋↑𝑏) ∈ 𝑆) → (𝜑 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
37 | 4, 7, 10, 13, 24, 36 | nn0ind 12076 | . 2 ⊢ (𝑌 ∈ ℕ0 → (𝜑 → (𝑋↑𝑌) ∈ 𝑆)) |
38 | 1, 37 | mpcom 38 | 1 ⊢ (𝜑 → (𝑋↑𝑌) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 ℕ0cn0 11896 ↑cexp 13428 SubRingcsubrg 19530 ℂfldccnfld 20544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-seq 13369 df-exp 13429 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-subg 18275 df-cmn 18907 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-subrg 19532 df-cnfld 20545 |
This theorem is referenced by: cnsrplycl 39765 |
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