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Theorem drnginvrl 20661

Description: Property of the multiplicative inverse in a division ring. (recid2 11920 analog). (Contributed by NM, 19-Apr-2014.)

**Hypotheses**
Ref	Expression
drnginvrl.b	⊢ 𝐵 = (Base‘𝑅)
drnginvrl.z	⊢ 0 = (0_g‘𝑅)
drnginvrl.t	⊢ · = (._r‘𝑅)
drnginvrl.u	⊢ 1 = (1_r‘𝑅)
drnginvrl.i	⊢ 𝐼 = (inv_r‘𝑅)

**Assertion**
Ref	Expression
drnginvrl	⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 )

**Proof of Theorem drnginvrl**
Step	Hyp	Ref	Expression
1		drnginvrl.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2725	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		drnginvrl.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
4	1, 2, 3	drngunit 20641	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
5		drngring 20643	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
6		drnginvrl.i	. . . . . 6 ⊢ 𝐼 = (inv_r‘𝑅)
7		drnginvrl.t	. . . . . 6 ⊢ · = (._r‘𝑅)
8		drnginvrl.u	. . . . . 6 ⊢ 1 = (1_r‘𝑅)
9	2, 6, 7, 8	unitlinv 20344	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Unit‘𝑅)) → ((𝐼‘𝑋) · 𝑋) = 1 )
10	9	ex 411	. . . 4 ⊢ (𝑅 ∈ Ring → (𝑋 ∈ (Unit‘𝑅) → ((𝐼‘𝑋) · 𝑋) = 1 ))
11	5, 10	syl 17	. . 3 ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ (Unit‘𝑅) → ((𝐼‘𝑋) · 𝑋) = 1 ))
12	4, 11	sylbird 259	. 2 ⊢ (𝑅 ∈ DivRing → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ))
13	12	3impib 1113	1 ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 ._rcmulr 17237 0_gc0g 17424 1_rcur 20133 Ringcrg 20185 Unitcui 20306 inv_rcinvr 20338 DivRingcdr 20636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-drng 20638

This theorem is referenced by: drnginvrld 20663 drngmul0or 20665 lvecvs0or 21008 lssvs0or 21010 lvecinv 21013 lspsnvs 21014 lspfixed 21028 lspsolv 21043 drngnidl 21150 matunitlindflem1 37220 lfl1 38672 eqlkr3 38703 lkrlsp 38704 tendolinv 40708 dochkr1 41081 dochkr1OLDN 41082 lclkrlem2m 41122 hdmapip1 41519 hgmapvvlem2 41527 prjspnfv01 42183

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