unitrinv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > unitrinv		Structured version Visualization version GIF version

Theorem unitrinv 20330

Description: A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)

**Hypotheses**
Ref	Expression
unitinvcl.1	⊢ 𝑈 = (Unit‘𝑅)
unitinvcl.2	⊢ 𝐼 = (inv_r‘𝑅)
unitinvcl.3	⊢ · = (._r‘𝑅)
unitinvcl.4	⊢ 1 = (1_r‘𝑅)

**Assertion**
Ref	Expression
unitrinv	⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )

**Proof of Theorem unitrinv**
Step	Hyp	Ref	Expression
1		unitinvcl.1	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
2		eqid 2736	. . . 4 ⊢ ((mulGrp‘𝑅) ↾_s 𝑈) = ((mulGrp‘𝑅) ↾_s 𝑈)
3	1, 2	unitgrp 20319	. . 3 ⊢ (𝑅 ∈ Ring → ((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp)
4	1, 2	unitgrpbas 20318	. . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾_s 𝑈))
5	1	fvexi 6848	. . . . 5 ⊢ 𝑈 ∈ V
6		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
7		unitinvcl.3	. . . . . . 7 ⊢ · = (._r‘𝑅)
8	6, 7	mgpplusg 20079	. . . . . 6 ⊢ · = (+_g‘(mulGrp‘𝑅))
9	2, 8	ressplusg 17211	. . . . 5 ⊢ (𝑈 ∈ V → · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
10	5, 9	ax-mp 5	. . . 4 ⊢ · = (+_g‘((mulGrp‘𝑅) ↾_s 𝑈))
11		eqid 2736	. . . 4 ⊢ (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈))
12		unitinvcl.2	. . . . 5 ⊢ 𝐼 = (inv_r‘𝑅)
13	1, 2, 12	invrfval 20325	. . . 4 ⊢ 𝐼 = (inv_g‘((mulGrp‘𝑅) ↾_s 𝑈))
14	4, 10, 11, 13	grprinv 18920	. . 3 ⊢ ((((mulGrp‘𝑅) ↾_s 𝑈) ∈ Grp ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
15	3, 14	sylan 580	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
16		unitinvcl.4	. . . 4 ⊢ 1 = (1_r‘𝑅)
17	1, 2, 16	unitgrpid 20321	. . 3 ⊢ (𝑅 ∈ Ring → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
18	17	adantr 480	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → 1 = (0_g‘((mulGrp‘𝑅) ↾_s 𝑈)))
19	15, 18	eqtr4d 2774	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ‘cfv 6492 (class class class)co 7358 ↾_s cress 17157 +_gcplusg 17177 ._rcmulr 17178 0_gc0g 17359 Grpcgrp 18863 mulGrpcmgp 20075 1_rcur 20116 Ringcrg 20168 Unitcui 20291 inv_rcinvr 20323

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-0g 17361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324

This theorem is referenced by: 1rinv 20331 0unit 20332 dvrid 20342 subrguss 20520 subrginv 20521 subrgunit 20523 drnginvrr 20690 matunit 22622 slesolinvbi 22625 nminvr 24613 nrginvrcnlem 24635 ply1divalg 26099 dchrn0 27217 dvrcan5 33318 dvdsruasso2 33467 fldhmf1 42344 invginvrid 48613

W3C validator