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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 2737 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 2 | domnexpgn0cl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | 1, 2 | mgpbas 20142 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
| 4 | domnexpgn0cl.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑅)) | |
| 5 | domnexpgn0cl.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Domn) | |
| 6 | domnring 20707 | . . . 4 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
| 7 | 1 | ringmgp 20236 | . . . 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 3963 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 12 | 3, 4, 8, 9, 11 | mulgnn0cld 19113 | . 2 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ 𝐵) |
| 13 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 0 → (𝑥 ↑ 𝑋) = (0 ↑ 𝑋)) | |
| 14 | 13 | neeq1d 3000 | . . . 4 ⊢ (𝑥 = 0 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (0 ↑ 𝑋) ≠ 0 )) |
| 15 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ↑ 𝑋) = (𝑦 ↑ 𝑋)) | |
| 16 | 15 | neeq1d 3000 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (𝑦 ↑ 𝑋) ≠ 0 )) |
| 17 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 ↑ 𝑋) = ((𝑦 + 1) ↑ 𝑋)) | |
| 18 | 17 | neeq1d 3000 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑥 ↑ 𝑋) ≠ 0 ↔ ((𝑦 + 1) ↑ 𝑋) ≠ 0 )) |
| 19 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑥 ↑ 𝑋) = (𝑁 ↑ 𝑋)) | |
| 20 | 19 | neeq1d 3000 | . . . 4 ⊢ (𝑥 = 𝑁 → ((𝑥 ↑ 𝑋) ≠ 0 ↔ (𝑁 ↑ 𝑋) ≠ 0 )) |
| 21 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 22 | 1, 21 | ringidval 20180 | . . . . . . 7 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅)) |
| 23 | 3, 22, 4 | mulg0 19092 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 ↑ 𝑋) = (1r‘𝑅)) |
| 24 | 11, 23 | syl 17 | . . . . 5 ⊢ (𝜑 → (0 ↑ 𝑋) = (1r‘𝑅)) |
| 25 | domnnzr 20706 | . . . . . 6 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
| 26 | domnexpgn0cl.0 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
| 27 | 21, 26 | nzrnz 20515 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ 0 ) |
| 28 | 5, 25, 27 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (1r‘𝑅) ≠ 0 ) |
| 29 | 24, 28 | eqnetrd 3008 | . . . 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 2737 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
| 34 | 1, 33 | mgpplusg 20141 | . . . . . . 7 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
| 35 | 3, 4, 34 | mulgnn0p1 19103 | . . . . . 6 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑦 + 1) ↑ 𝑋) = ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋)) |
| 36 | 30, 31, 32, 35 | syl3anc 1373 | . . . . 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 19113 | . . . . . 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 4790 | . . . . . . . 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 20709 | . . . . . 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 1381 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → ((𝑦 ↑ 𝑋)(.r‘𝑅)𝑋) ≠ 0 ) |
| 45 | 36, 44 | eqnetrd 3008 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ (𝑦 ↑ 𝑋) ≠ 0 ) → ((𝑦 + 1) ↑ 𝑋) ≠ 0 ) |
| 46 | 14, 16, 18, 20, 29, 45 | nn0indd 12715 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (𝑁 ↑ 𝑋) ≠ 0 ) |
| 47 | 9, 46 | mpdan 687 | . 2 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ≠ 0 ) |
| 48 | 12, 47 | eldifsnd 4787 | 1 ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐵 ∖ { 0 })) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 0gc0g 17484 Mndcmnd 18747 .gcmg 19085 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 NzRingcnzr 20512 Domncdomn 20692 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-nzr 20513 df-domn 20695 |
| This theorem is referenced by: fidomncyc 42545 |
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