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 6557 | . . . . 5 ⊢ (𝑎 = 0 → (𝑋↑𝑎) = (𝑋↑0)) | |
3 | 2 | eleq1d 2672 | . . . 4 ⊢ (𝑎 = 0 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑0) ∈ 𝑆)) |
4 | 3 | imbi2d 329 | . . 3 ⊢ (𝑎 = 0 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑0) ∈ 𝑆))) |
5 | oveq2 6557 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑋↑𝑎) = (𝑋↑𝑏)) | |
6 | 5 | eleq1d 2672 | . . . 4 ⊢ (𝑎 = 𝑏 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑏) ∈ 𝑆)) |
7 | 6 | imbi2d 329 | . . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑏) ∈ 𝑆))) |
8 | oveq2 6557 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝑋↑𝑎) = (𝑋↑(𝑏 + 1))) | |
9 | 8 | eleq1d 2672 | . . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑(𝑏 + 1)) ∈ 𝑆)) |
10 | 9 | imbi2d 329 | . . 3 ⊢ (𝑎 = (𝑏 + 1) → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑(𝑏 + 1)) ∈ 𝑆))) |
11 | oveq2 6557 | . . . . 5 ⊢ (𝑎 = 𝑌 → (𝑋↑𝑎) = (𝑋↑𝑌)) | |
12 | 11 | eleq1d 2672 | . . . 4 ⊢ (𝑎 = 𝑌 → ((𝑋↑𝑎) ∈ 𝑆 ↔ (𝑋↑𝑌) ∈ 𝑆)) |
13 | 12 | imbi2d 329 | . . 3 ⊢ (𝑎 = 𝑌 → ((𝜑 → (𝑋↑𝑎) ∈ 𝑆) ↔ (𝜑 → (𝑋↑𝑌) ∈ 𝑆))) |
14 | cnsrexpcl.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
15 | cnfldbas 19571 | . . . . . . . 8 ⊢ ℂ = (Base‘ℂfld) | |
16 | 15 | subrgss 18604 | . . . . . . 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 3569 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
20 | 19 | exp0d 12864 | . . . 4 ⊢ (𝜑 → (𝑋↑0) = 1) |
21 | cnfld1 19590 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 21 | subrg1cl 18611 | . . . . 5 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑆) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) |
24 | 20, 23 | eqeltrd 2688 | . . 3 ⊢ (𝜑 → (𝑋↑0) ∈ 𝑆) |
25 | 19 | 3ad2ant2 1076 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ ℂ) |
26 | simp1 1054 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑏 ∈ ℕ0) | |
27 | 25, 26 | expp1d 12871 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) = ((𝑋↑𝑏) · 𝑋)) |
28 | 14 | 3ad2ant2 1076 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑆 ∈ (SubRing‘ℂfld)) |
29 | simp3 1056 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑𝑏) ∈ 𝑆) | |
30 | 18 | 3ad2ant2 1076 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
31 | cnfldmul 19573 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
32 | 31 | subrgmcl 18615 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑋↑𝑏) ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
33 | 28, 29, 30, 32 | syl3anc 1318 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
34 | 27, 33 | eqeltrd 2688 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) ∈ 𝑆) |
35 | 34 | 3exp 1256 | . . . 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 11348 | . 2 ⊢ (𝑌 ∈ ℕ0 → (𝜑 → (𝑋↑𝑌) ∈ 𝑆)) |
38 | 1, 37 | mpcom 37 | 1 ⊢ (𝜑 → (𝑋↑𝑌) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℕ0cn0 11169 ↑cexp 12722 SubRingcsubrg 18599 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-seq 12664 df-exp 12723 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-cnfld 19568 |
This theorem is referenced by: cnsrplycl 36756 |
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