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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 7420 | . . . . 5 ⊢ (𝑎 = 0 → (𝑋↑𝑎) = (𝑋↑0)) | |
3 | 2 | eleq1d 2817 | . . . 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 7420 | . . . . 5 ⊢ (𝑎 = 𝑏 → (𝑋↑𝑎) = (𝑋↑𝑏)) | |
6 | 5 | eleq1d 2817 | . . . 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 7420 | . . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝑋↑𝑎) = (𝑋↑(𝑏 + 1))) | |
9 | 8 | eleq1d 2817 | . . . 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 7420 | . . . . 5 ⊢ (𝑎 = 𝑌 → (𝑋↑𝑎) = (𝑋↑𝑌)) | |
12 | 11 | eleq1d 2817 | . . . 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 21236 | . . . . . . . 8 ⊢ ℂ = (Base‘ℂfld) | |
16 | 15 | subrgss 20470 | . . . . . . 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 3983 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
20 | 19 | exp0d 14112 | . . . 4 ⊢ (𝜑 → (𝑋↑0) = 1) |
21 | cnfld1 21258 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
22 | 21 | subrg1cl 20478 | . . . . 5 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 1 ∈ 𝑆) |
23 | 14, 22 | syl 17 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝑆) |
24 | 20, 23 | eqeltrd 2832 | . . 3 ⊢ (𝜑 → (𝑋↑0) ∈ 𝑆) |
25 | 19 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ ℂ) |
26 | simp1 1135 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑏 ∈ ℕ0) | |
27 | 25, 26 | expp1d 14119 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) = ((𝑋↑𝑏) · 𝑋)) |
28 | 14 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑆 ∈ (SubRing‘ℂfld)) |
29 | simp3 1137 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑𝑏) ∈ 𝑆) | |
30 | 18 | 3ad2ant2 1133 | . . . . . . 7 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → 𝑋 ∈ 𝑆) |
31 | cnfldmul 21238 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
32 | 31 | subrgmcl 20482 | . . . . . . 7 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ (𝑋↑𝑏) ∈ 𝑆 ∧ 𝑋 ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
33 | 28, 29, 30, 32 | syl3anc 1370 | . . . . . 6 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → ((𝑋↑𝑏) · 𝑋) ∈ 𝑆) |
34 | 27, 33 | eqeltrd 2832 | . . . . 5 ⊢ ((𝑏 ∈ ℕ0 ∧ 𝜑 ∧ (𝑋↑𝑏) ∈ 𝑆) → (𝑋↑(𝑏 + 1)) ∈ 𝑆) |
35 | 34 | 3exp 1118 | . . . 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 12664 | . 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 1086 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 ℕ0cn0 12479 ↑cexp 14034 SubRingcsubrg 20465 ℂfldccnfld 21232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-seq 13974 df-exp 14035 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-subrng 20442 df-subrg 20467 df-cnfld 21233 |
This theorem is referenced by: cnsrplycl 42371 |
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