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Mirrors > Home > MPE Home > Th. List > Mathboxes > domnexpgn0cl | Structured version Visualization version GIF version |
Description: In a domain, a (nonnegative) power of a nonzero element is nonzero. (Contributed by SN, 6-Jul-2024.) |
Ref | Expression |
---|---|
domnexpgn0cl.b | ⊢ 𝐵 = (Base‘𝑅) |
domnexpgn0cl.0 | ⊢ 0 = (0g‘𝑅) |
domnexpgn0cl.e | ⊢ ↑ = (.g‘(mulGrp‘𝑅)) |
domnexpgn0cl.r | ⊢ (𝜑 → 𝑅 ∈ Domn) |
domnexpgn0cl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
domnexpgn0cl.x | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) |
Ref | Expression |
---|---|
domnexpgn0cl | ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐵 ∖ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | domnexpgn0cl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 20157 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
4 | domnexpgn0cl.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑅)) | |
5 | domnexpgn0cl.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
6 | domnring 20723 | . . . 4 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
7 | 1 | ringmgp 20256 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ Mnd) |
9 | domnexpgn0cl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
10 | domnexpgn0cl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 })) | |
11 | 10 | eldifad 3974 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
12 | 3, 4, 8, 9, 11 | mulgnn0cld 19125 | . 2 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ 𝐵) |
13 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ 𝑋) = (0 ↑ 𝑋)) | |
14 | 13 | neeq1d 2997 | . . . 4 ⊢ (𝑥 = 0 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (0 ↑ 𝑋) ≠ 0 )) |
15 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ↑ 𝑋) = (𝑦 ↑ 𝑋)) | |
16 | 15 | neeq1d 2997 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (𝑦 ↑ 𝑋) ≠ 0 )) |
17 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ↑ 𝑋) = ((𝑦 + 1) ↑ 𝑋)) | |
18 | 17 | neeq1d 2997 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 ↑ 𝑋) ≠ 0 ↔ ((𝑦 + 1) ↑ 𝑋) ≠ 0 )) |
19 | oveq1 7437 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ 𝑋) = (𝑁 ↑ 𝑋)) | |
20 | 19 | neeq1d 2997 | . . . 4 ⊢ (𝑥 = 𝑁 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (𝑁 ↑ 𝑋) ≠ 0 )) |
21 | eqid 2734 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
22 | 1, 21 | ringidval 20200 | . . . . . . 7 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
23 | 3, 22, 4 | mulg0 19104 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 ↑ 𝑋) = (1r‘𝑅)) |
24 | 11, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (0 ↑ 𝑋) = (1r‘𝑅)) |
25 | domnnzr 20722 | . . . . . 6 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
26 | domnexpgn0cl.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
27 | 21, 26 | nzrnz 20531 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
28 | 5, 25, 27 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ 0 ) |
29 | 24, 28 | eqnetrd 3005 | . . . 4 ⊢ (𝜑 → (0 ↑ 𝑋) ≠ 0 ) |
30 | 8 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → (mulGrp‘𝑅) ∈ Mnd) |
31 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → 𝑦 ∈ ℕ0) | |
32 | 11 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → 𝑋 ∈ 𝐵) |
33 | eqid 2734 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
34 | 1, 33 | mgpplusg 20155 | . . . . . . 7 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
35 | 3, 4, 34 | mulgnn0p1 19115 | . . . . . 6 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) ↑ 𝑋) = ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋)) |
36 | 30, 31, 32, 35 | syl3anc 1370 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → ((𝑦 + 1) ↑ 𝑋) = ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋)) |
37 | 5 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → 𝑅 ∈ Domn) |
38 | 3, 4, 30, 31, 32 | mulgnn0cld 19125 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → (𝑦 ↑ 𝑋) ∈ 𝐵) |
39 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → (𝑦 ↑ 𝑋) ≠ 0 ) | |
40 | eldifsni 4794 | . . . . . . . 8 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) → 𝑋 ≠ 0 ) | |
41 | 10, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 0 ) |
42 | 41 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → 𝑋 ≠ 0 ) |
43 | 2, 33, 26 | domnmuln0 20725 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ ((𝑦 ↑ 𝑋) ∈ 𝐵 ∧ (𝑦 ↑ 𝑋) ≠ 0 ) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋) ≠ 0 ) |
44 | 37, 38, 39, 32, 42, 43 | syl122anc 1378 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋) ≠ 0 ) |
45 | 36, 44 | eqnetrd 3005 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → ((𝑦 + 1) ↑ 𝑋) ≠ 0 ) |
46 | 14, 16, 18, 20, 29, 45 | nn0indd 12712 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 ↑ 𝑋) ≠ 0 ) |
47 | 9, 46 | mpdan 687 | . 2 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ≠ 0 ) |
48 | 12, 47 | eldifsnd 4791 | 1 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐵 ∖ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 ‘cfv 6562 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 ℕ0cn0 12523 Basecbs 17244 .rcmulr 17298 0gc0g 17485 Mndcmnd 18759 .gcmg 19097 mulGrpcmgp 20151 1rcur 20198 Ringcrg 20250 NzRingcnzr 20528 Domncdomn 20708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-seq 14039 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-mulg 19098 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-nzr 20529 df-domn 20711 |
This theorem is referenced by: fidomncyc 42521 |
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