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 6658 | . . . . 5 ⊢ (𝑎 = 0 → (𝑋↑𝑎) = (𝑋↑0)) | |
3 | 2 | eleq1d 2686 | . . . 4 ⊢ (𝑎 = 0 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑0) ∈ 𝑆)) |
4 | 3 | imbi2d 330 | . . 3 ⊢ (𝑎 = 0 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑0) ∈ 𝑆))) |
5 | oveq2 6658 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑋↑𝑎) = (𝑋↑𝑏)) | |
6 | 5 | eleq1d 2686 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑏) ∈ 𝑆)) |
7 | 6 | imbi2d 330 | . . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑏) ∈ 𝑆))) |
8 | oveq2 6658 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝑋↑𝑎) = (𝑋↑(𝑏 + 1))) | |
9 | 8 | eleq1d 2686 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑(𝑏 + 1)) ∈ 𝑆)) |
10 | 9 | imbi2d 330 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
11 | oveq2 6658 | . . . . 5 ⊢ (𝑎 = 𝑌 → (𝑋↑𝑎) = (𝑋↑𝑌)) | |
12 | 11 | eleq1d 2686 | . . . 4 ⊢ (𝑎 = 𝑌 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑌) ∈ 𝑆)) |
13 | 12 | imbi2d 330 | . . 3 ⊢ (𝑎 = 𝑌 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑌) ∈ 𝑆))) |
14 | cnsrexpcl.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
15 | cnfldbas 19750 | . . . . . . . 8 ⊢ ℂ = (Base‘ℂfld) | |
16 | 15 | subrgss 18781 | . . . . . . 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 3604 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
20 | 19 | exp0d 13002 | . . . 4 ⊢ (𝜑 → (𝑋↑0) = 1) |
21 | cnfld1 19771 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 21 | subrg1cl 18788 | . . . . 5 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑆) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) |
24 | 20, 23 | eqeltrd 2701 | . . 3 ⊢ (𝜑 → (𝑋↑0) ∈ 𝑆) |
25 | 19 | 3ad2ant2 1083 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ ℂ) |
26 | simp1 1061 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑏 ∈ ℕ0) | |
27 | 25, 26 | expp1d 13009 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) = ((𝑋↑𝑏) · 𝑋)) |
28 | 14 | 3ad2ant2 1083 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑆 ∈ (SubRing‘ℂfld)) |
29 | simp3 1063 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑𝑏) ∈ 𝑆) | |
30 | 18 | 3ad2ant2 1083 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
31 | cnfldmul 19752 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
32 | 31 | subrgmcl 18792 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑋↑𝑏) ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
33 | 28, 29, 30, 32 | syl3anc 1326 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
34 | 27, 33 | eqeltrd 2701 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) ∈ 𝑆) |
35 | 34 | 3exp 1264 | . . . 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 11472 | . 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 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℕ0cn0 11292 ↑cexp 12860 SubRingcsubrg 18776 ℂfldccnfld 19746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-exp 12861 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-cnfld 19747 |
This theorem is referenced by: cnsrplycl 37737 |
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