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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 7365 | . . . . 5 ⊢ (𝑎 = 0 → (𝑋↑𝑎) = (𝑋↑0)) | |
3 | 2 | eleq1d 2822 | . . . 4 ⊢ (𝑎 = 0 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑0) ∈ 𝑆)) |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑎 = 0 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑0) ∈ 𝑆))) |
5 | oveq2 7365 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑋↑𝑎) = (𝑋↑𝑏)) | |
6 | 5 | eleq1d 2822 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑏) ∈ 𝑆)) |
7 | 6 | imbi2d 340 | . . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑏) ∈ 𝑆))) |
8 | oveq2 7365 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝑋↑𝑎) = (𝑋↑(𝑏 + 1))) | |
9 | 8 | eleq1d 2822 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑(𝑏 + 1)) ∈ 𝑆)) |
10 | 9 | imbi2d 340 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
11 | oveq2 7365 | . . . . 5 ⊢ (𝑎 = 𝑌 → (𝑋↑𝑎) = (𝑋↑𝑌)) | |
12 | 11 | eleq1d 2822 | . . . 4 ⊢ (𝑎 = 𝑌 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑌) ∈ 𝑆)) |
13 | 12 | imbi2d 340 | . . 3 ⊢ (𝑎 = 𝑌 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑌) ∈ 𝑆))) |
14 | cnsrexpcl.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
15 | cnfldbas 20800 | . . . . . . . 8 ⊢ ℂ = (Base‘ℂfld) | |
16 | 15 | subrgss 20223 | . . . . . . 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 3945 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
20 | 19 | exp0d 14045 | . . . 4 ⊢ (𝜑 → (𝑋↑0) = 1) |
21 | cnfld1 20822 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 21 | subrg1cl 20230 | . . . . 5 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑆) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) |
24 | 20, 23 | eqeltrd 2838 | . . 3 ⊢ (𝜑 → (𝑋↑0) ∈ 𝑆) |
25 | 19 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ ℂ) |
26 | simp1 1136 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑏 ∈ ℕ0) | |
27 | 25, 26 | expp1d 14052 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) = ((𝑋↑𝑏) · 𝑋)) |
28 | 14 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑆 ∈ (SubRing‘ℂfld)) |
29 | simp3 1138 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑𝑏) ∈ 𝑆) | |
30 | 18 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
31 | cnfldmul 20802 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
32 | 31 | subrgmcl 20234 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑋↑𝑏) ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
33 | 28, 29, 30, 32 | syl3anc 1371 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
34 | 27, 33 | eqeltrd 2838 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) ∈ 𝑆) |
35 | 34 | 3exp 1119 | . . . 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 12598 | . 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 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ℕ0cn0 12413 ↑cexp 13967 SubRingcsubrg 20218 ℂfldccnfld 20796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-seq 13907 df-exp 13968 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-grp 18751 df-subg 18925 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-cnfld 20797 |
This theorem is referenced by: cnsrplycl 41480 |
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