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Mirrors > Home > MPE Home > Th. List > drnginvrn0 | Structured version Visualization version GIF version |
Description: The multiplicative inverse in a division ring is nonzero. (recne0 11402 analog). (Contributed by NM, 19-Apr-2014.) |
Ref | Expression |
---|---|
invrcl.b | ⊢ 𝐵 = (Base‘𝑅) |
invrcl.z | ⊢ 0 = (0g‘𝑅) |
invrcl.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
drnginvrn0 | ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngring 19641 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
2 | eqid 2739 | . . . . . . 7 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | invrcl.i | . . . . . . 7 ⊢ 𝐼 = (invr‘𝑅) | |
4 | 2, 3 | unitinvcl 19559 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → (𝐼‘𝑋) ∈ (Unit‘𝑅)) |
5 | 4 | ex 416 | . . . . 5 ⊢ (𝑅 ∈ Ring → (𝑋 ∈ (Unit‘𝑅) → (𝐼‘𝑋) ∈ (Unit‘𝑅))) |
6 | 1, 5 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) → (𝐼‘𝑋) ∈ (Unit‘𝑅))) |
7 | invrcl.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
8 | invrcl.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
9 | 7, 2, 8 | drngunit 19639 | . . . 4 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) |
10 | 7, 2, 8 | drngunit 19639 | . . . 4 ⊢ (𝑅 ∈ DivRing → ((𝐼‘𝑋) ∈ (Unit‘𝑅) ↔ ((𝐼‘𝑋) ∈ 𝐵 ∧ (𝐼‘𝑋) ≠ 0 ))) |
11 | 6, 9, 10 | 3imtr3d 296 | . . 3 ⊢ (𝑅 ∈ DivRing → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) ∈ 𝐵 ∧ (𝐼‘𝑋) ≠ 0 ))) |
12 | 11 | 3impib 1117 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) ∈ 𝐵 ∧ (𝐼‘𝑋) ≠ 0 )) |
13 | 12 | simprd 499 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ‘cfv 6350 Basecbs 16599 0gc0g 16829 Ringcrg 19429 Unitcui 19524 invrcinvr 19556 DivRingcdr 19634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-tpos 7934 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-0g 16831 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-grp 18235 df-minusg 18236 df-mgp 19372 df-ur 19384 df-ring 19431 df-oppr 19508 df-dvdsr 19526 df-unit 19527 df-invr 19557 df-drng 19636 |
This theorem is referenced by: lspfixed 20032 extdg1id 31323 tendoinvcl 38774 dochkr1 39148 lcfrlem31 39243 mapdpglem18 39359 mapdpglem22 39363 hgmapvvlem2 39594 drnginvrn0d 39872 prjspner01 40080 |
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